The Derived Subgroups of Sylow 2-Subgroups of the Alternating Group, Commutator Width of Wreath Product of Groups

Abstract

The structure of the commutator subgroup of Sylow 2-subgroups of an alternating group $A_{2^k}$ is determined. This work continues the previous investigations of me, where minimal generating sets for Sylow 2-subgroups of alternating groups were constructed. Here we study the commutator subgroup of these groups. The minimal generating set of the commutator subgroup of $A_{2^k}$ is constructed. It is shown that ${\left(Sy{l}_2{A}_{2^k}\right)}^2=Sy{l}_2^{\prime }{A}_{2^k},k>2$. It serves to solve quadratic equations in this group, as were solved by Lysenok I. in the Grigorchuk group. It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups $C_{p_i},p_i\in \mathbb{N}$ equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. Upper bounds for the commutator width $\left(cw\right(G\left)\right)$ of a wreath product of groups are presented in this paper. A presentation in form of wreath recursion of Sylow 2-subgroups $Sy{l}_2\left(A_{2^k}\right)$ of $A_{2^k}$ is introduced. As a result, a short proof that the commutator width is equal to 1 for Sylow 2-subgroups of alternating group $A_{2^k}$, where $k>2$, the permutation group $S_{2^k}$, as well as Sylow p-subgroups of $Sy{l}_2{A}_{p^k}$ as well as $Sy{l}_2{S}_{p^k}$) are equal to 1 was obtained. A commutator width of permutational wreath product $B\wr C_n$ is investigated. An upper bound of the commutator width of permutational wreath product $B\wr C_n$ for an arbitrary group B is found. The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroup of the alternating group is found. The proofs were assisted by the computer algebra system GAP.

1. Introduction

The object of our study is the commutatorwidth [1] of Sylow 2-subgroups of alternating group $A_{2^k}$. As an intermediate goal, we have a structural description of the derived subgroup of this subgroup. The commutator width of G is the minimal n such that for arbitrary $g\in [G,G]$ there exist elements $x_1,\dots ,x_n,y_1,\dots ,y_n$ in G such that $g=[x_1,y_1]\dots [x_n,y_n]$.

Our study of the width of the commutator is somewhat similar to the study of equations in simple matrix groups [2], and is also associated with verbal subgroups. Additionally, in related work [3], it was established that the commutator width of the first Grigorchuk group is 2.

Commutator width of groups, and of elements, has proven to be an important group property, in particular via its connections with stable commutator length and bounded cohomology [4,5]. It is also related to solvability of quadratic equations in groups [6]: a group G has commutator width $\le n$ if and only if the equation $[X_1,X_2]\dots [X_{2n-1},X_{2n}]g=1$ is solvable for all $g\in G^{\prime }$.

As it is well known, the first example of a group G with commutator width greater than 1 ($cw\left(G\right)>1$) was given by Fite [7]. The smallest finite examples of such groups are groups of order 96; there are two of them, nonisomorphic to each other, which were given by Guralnick [8].

We obtain an upper bound for commutator width of wreath product $C_n\wr B$, where $C_n$ is cyclic group of order n, in terms of the commutator width $cw\left(B\right)$ of passive group B. A form of commutators of wreath product $A\wr B$ was briefly considered in [9]. The form of commutator presentation [9] is proposed by us as wreath recursion [10], and the commutator width of it was studied. We imposed a weaker condition on the presentation of wreath product commutator than was proposed by J. Meldrum.

In this paper we continue investigations started in [11,12,13,14,15,16,17]. We find a minimal generating set and the structure for commutator subgroup of $Sy{l}_2{A}_{2^k}$.

Research of commutator-group serves the decision of inclusion problem [18] for elements of $Sy{l}_2{A}_{2^k}$ in its derived subgroup ${\left(Sy{l}_2{A}_{2^k}\right)}^{\prime }$. Knowledge of the method for solving the of inclusion problem in a subgroup H facilitates the solution of the problem of finding the conjugate elements in the whole group (conjugacy search problem) [19]. Because by the characterization of the conjugated elements g and $h^{-1}gh$, we can determine which subgroups they belong to and which do not belong.

It is known that the commutator width of iterated wreath products of nonabelian finite simple groups is bounded by an absolute constant [7,20]. But it has not been proven that commutator subgroup of $\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}$ consists of commutators. We generalize the passive group of this wreath product to any group B instead of only wreath product of cyclic groups and obtain an exact commutator width.

Additionally, we are going to prove that the commutator width of Sylow p-subgroups of symmetric and alternating groups for $p\ge 2$ is 1.

2. Preliminaries

Let G be a group acting (from the right) by permutations on a set X and let H be an arbitrary group. Then the (permutational) wreath product $H\wr G$ is the semidirect product $H^X⋋G$, where G acts on the direct power $H^X$ by the respective permutations of the direct factors. The cyclic group $C_p$ or $(C_p,X)$ is equipped with a natural action by the left shift on $X=\{1,...,p\}$, $p\in \mathbb{N}$. It is well known that a wreath product of permutation groups is associative construction [9].

The multiplication rule of automorphisms g and h, which are presented in form of the wreath recursion [21] $g=(g_{\left(1\right)},g_{\left(2\right)},\dots ,g_{\left(d\right)}){\sigma }_g,h=(h_{\left(1\right)},h_{\left(2\right)},\dots ,h_{\left(d\right)}){\sigma }_h,$ is given by the formula:

$$g·h=(g_{\left(1\right)}{h}_{\left({\sigma }_g\left(1\right)\right)},g_{\left(2\right)}{h}_{\left({\sigma }_g\left(2\right)\right)},\dots ,g_{\left(d\right)}{h}_{\left({\sigma }_g\left(d\right)\right)}){\sigma }_g{\sigma }_h.$$

We define $\sigma$ as $(1,2,\dots ,p)$ where p is defined by context.

The set $X^{*}$ is naturally a vertex set of a regular rooted tree; i.e., a connected graph without cycles and a designated vertex $v_0$ called the root, in which two words are connected by an edge if and only if they are of form v and $vx$, where $v\in X^{*}$, $x\in X$. The set $X^n\subset X^{*}$ is called the n-th level of the tree $X^{*}$ and $X^0=\left\{v_0\right\}$. We denote by $v_{ji}$ the vertex of $X^j$, which has the number i, where $1\le i\le X^{2^j}$ and the numeration starts from 1. Note that the unique vertex $v_{k,i}$ corresponds to the unique word v in alphabet X. For every automorphism $g\in Aut{X}^{*}$ and every word $v\in X^{*}$ determine the section (state) $g_{\left(v\right)}\in Aut{X}^{*}$ of g at v by the rule: $g_{\left(v\right)}\left(x\right)=y$ for $x,y\in X^{*}$ if and only if $g\left(vx\right)=g\left(v\right)y$. The subtree of $X^{*}$ induced by the set of vertices ${\cup }_{i=0}^k{X}^i$ is denoted by $X^{\left[k\right]}$. The restriction of the action of an automorphism $g\in Aut{X}^{*}$ to the subtree $X^{\left[l\right]}$ is denoted by $g_{\left(v\right)}{|}_{X^{\left[l\right]}}$. The restriction $g_{\left(v_{ij}\right)}{|}_{X^{\left[1\right]}}$ is called the vertex permutation (v.p.) of g at a vertex $v_{ij}$ and denoted by $g_{ij}$. For example, if $\left|X\right|=2$ then we just have to distinguish active vertices; i.e. the vertices for which $g_{ij}$ is non-trivial [21].

We label every vertex of $X^l,0\le l<k$ by 0 or 1 depending on the action of v.p. on it. The resulting vertex-labeled regular tree is an element of $Aut{X}^{\left[k\right]}$. All undeclared terms are from [22,23,24].

Let us fix some notation. For convenience the commutator of two group elements a and b is denoted by $[a,b]=ab{a}^{-1}{b}^{-1},$ conjugation by an element b we denote by

$$a^b=ba{b}^{-1}.$$

We define $G_k$ and $B_k$ recursively; i.e.,

$$\begin{array}{cc}\hfill B_1& =C_2,B_k=B_{k-1}\wr C_2\mathrm{for}k>1,\hfill \\ \hfill G_1& =\langle e\rangle ,G_k=\{(g_1,g_2)\pi \in B_k\mid g_1{g}_2\in G_{k-1}\}\mathrm{for}k>1.\hfill \end{array}$$

Note that $B_k=\stackrel{k}{\underset{\mathrm{i}=1}{\wr }}{C}_2$.

The commutator length of an element g of a derived subgroup of a group G, is the minimal n such that there exist elements $x_1,\dots ,x_n,y_1,\dots ,y_n$ in G such that $g=[x_1,y_1]\dots [x_n,y_n]$. The commutator length of the identity element is 0. Let clG(g) denotes the commutator length of an element g of a group G. The commutator width of a group G is the maximum of clG(g) of the elements of its derived subgroup $[G,G]$. We denote by $d\left(G\right)$ the minimal number of generators of the group G.

3. Commutator Width of Sylow 2-Subgroups of $A_{2^k}$ and $S_{2^k}$

The the following lemma improves the result stated as Corollary 4.9 in of [9]. Our proof uses arguments similar to those of [9].

Lemma 1.

An element of form $(r_1,\dots ,r_{p-1},r_p)\in W^{\prime }={(B\wr C_p)}^{\prime }$ iff product of all $r_i$ (in any order) belongs to $B^{\prime }$, where $p\in \mathrm{N}$, $p\ge 2$.

Proof. 

More details of our argument may be given as follows. If we multiply elements from a tuple $(r_1,\dots ,r_{p-1},r_p)=w$, where $r_i=h_i{g}_{a\left(i\right)}{h}_{ab\left(i\right)}^{-1}{g}_{ab{a}^{-1}\left(i\right)}^{-1}$, $h_i,g_i\in B$ and $a,b\in C_p$, then we get a product

$$x=\stackrel{p}{\prod _{\mathrm{i}=1}}{r}_i=\prod _{i=1}^p{h}_i{g}_{a\left(i\right)}{h}_{ab\left(i\right)}^{-1}{g}_{ab{a}^{-1}\left(i\right)}^{-1}\in B^{\prime },$$

(1)

where x is a product of appropriate commutators. Therefore, we can write $r_p=r_{p-1}^{-1}\dots r_1^{-1}x$. We can rewrite element $x\in B^{\prime }$ as the product $x={\displaystyle \prod _{j=1}^m}[h_j,g_j]$, $m\le cw\left(B\right)$.

Note that we impose a weaker condition on the product of all $r_i$ which belongs to $B^{\prime }$ than in Definition 4.5 of form $P\left(L\right)$ in [9], where the product of all $r_i$ belongs to a subgroup L of B such that $L>B^{\prime }$.

In more detail, deducing of our representation construct can be reported in the following way. If we multiply elements having form of a tuple $(r_1,\dots ,r_{p-1},r_p)$, where $r_i=h_i{g}_{a\left(i\right)}{h}_{ab\left(i\right)}^{-1}{g}_{ab{a}^{-1}\left(i\right)}^{-1}$, $h_i,g_i\in B$ and $a,b\in C_p$, then we obtain a product

$$\stackrel{p}{\prod _{\mathrm{i}=1}}{r}_i=\prod _{i=1}^p{h}_i{g}_{a\left(i\right)}{h}_{ab\left(i\right)}^{-1}{g}_{ab{a}^{-1}\left(i\right)}^{-1}\in B^{\prime }.$$

(2)

Note that if we rearrange elements in (1) as $h_1{h}_1^{-1}{g}_1{g}_2^{-1}{h}_2{h}_2^{-1}{g}_1{g}_2^{-1}...h_p{h}_p^{-1}{g}_p{g}_p^{-1}$ then by the reason of such permutations we obtain a product of appropriate commutators. Therefore, the following equality holds

$$\prod _{i=1}^p{h}_i{g}_{a\left(i\right)}{h}_{ab\left(i\right)}^{-1}{g}_{ab{a}^{-1}\left(i\right)}^{-1}=\prod _{i=1}^p{h}_i{g}_i{h}_i^{-1}{g}_i^{-1}{x}_0=\prod _{i=1}^p{h}_i{h}_i^{-1}{g}_i{g}_i^{-1}x\in B^{\prime },$$

(3)

where $x_0,x$ are the products of appropriate commutators. Therefore,

$$\begin{array}{c}\hfill (r_1,\dots ,r_{p-1},r_p)\in W^{\prime }\mathrm{iff}{r}_{p-1}·\dots ·r_1·r_p=x\in B^{\prime }.\end{array}$$

(4)

Thus, one element from states of wreath recursion $(r_1,\dots ,r_{p-1},r_p)$ depends on rest of $r_i$. This implies that the product ${\displaystyle \prod _{j=1}^p}{r}_j$ for an arbitrary sequence ${\left\{r_j\right\}}_{j=1}^p$ belongs to $B^{\prime }$. Thus, $r_p$ can be expressed as:

$$\begin{array}{c}\hfill r_p=r_1^{-1}·\dots ·r_{p-1}^{-1}x.\end{array}$$

Denote a j-th tuple consisting of wreath recursion elements by $(r_{j_1},r_{j_2},...,r_{j_p})$. The fact that the set of forms $(r_1,\dots ,r_{p-1},r_p)\in W={(B\wr C_p)}^{\prime }$ is closed under multiplication follows from the identity follows from

$$\begin{array}{c}\hfill \prod _{j=1}^k(r_{j1}\dots r_{jp-1}{r}_{jp})=\prod _{j=1}^k\prod _{i=1}^p{r}_{j_i}=R_1{R}_2...R_k\in B^{\prime },\end{array}$$

(5)

where $r_{ji}$ is i-th element of the tuple number j, $R_j={\displaystyle \prod _{i=1}^p}{r}_{ji},1\le j\le k$. As it was shown above $R_j={\displaystyle \prod _{i=1}^{p-1}}{r}_{ji}\in B^{\prime }$. Therefore, the product (5) of $R_j$, $j\in \{1,...,k\}$ which is similar to the product mentioned in [9], has the property $R_1{R}_2...R_k\in B^{\prime }$ too, because of $B^{\prime }$ is subgroup. Thus, we get a product of form (1) and the similar reasoning as above is applicable.

Let us prove the sufficiency condition. If the set K of elements satisfying the condition of this theorem, that all products of all $r_i$, where every i occurs in this form once, belong to $B^{\prime }$. Then using the elements of the form

$$(r_1,e,\dots ,e,r_1^{-1}), \dots , (e,e,\dots ,e,r_i,e,r_i^{-1}), \dots , (e,e, \dots ,e,r_{p-1},r_{p-1}^{-1}), (e,e,\dots ,e,r_1{r}_2·\dots ·r_{p-1})$$

we can express any elements of the form $(r_1,\dots ,r_{p-1},r_p)\in W={(B\wr C_p)}^{\prime }$. We need to prove that in such a way we can express all element from W and only elements of W. All elements of W can be generated by elements of K since $r_i$, $i<p$ are arbitrary and the fact that equality (1) holds, so $r_p$ is well determined. □

Lemma 2.

Assume a group B and an integer $p\ge 2$. If $w\in {(B\wr C_p)}^{\prime }$ then w can be represented as the following wreath recursion

$$\begin{array}{c}\hfill w=(r_1,r_2,\dots ,r_{p-1},r_1^{-1}\dots r_{p-1}^{-1}\prod _{j=1}^k[f_j,g_j]),\end{array}$$

where $r_1,\dots ,r_{p-1},f_j,g_j\in B$ and $k\le cw\left(B\right)$.

Proof. 

According to Lemma 1 we have the following wreath recursion

$$\begin{array}{c}\hfill w=(r_1,r_2,\dots ,r_{p-1},r_p),\end{array}$$

where $r_i\in B$ and $r_{p-1}{r}_{p-2}\dots r_2{r}_1{r}_p=x\in B^{\prime }$. Therefore, we can write $r_p=r_1^{-1}\dots r_{p-1}^{-1}x$. We can also rewrite an element $x\in B^{\prime }$ as a product of commutators $x={\displaystyle \prod _{j=1}^k}[f_j,g_j]$ where $k\le cw\left(B\right)$. □

Lemma 3.

For any group B and integer $p\ge 2$, suppose $w\in {(B\wr C_p)}^{\prime }$ is defined by the following wreath recursion:

$$\begin{array}{c}\hfill w=(r_1,r_2,\dots ,r_{p-1},r_1^{-1}\dots r_{p-1}^{-1}[f,g]),\end{array}$$

where $r_1,\dots ,r_{p-1},f,g\in B$. Then we can represent w as the following commutator

$$\begin{array}{c}\hfill w=[(a_{1,1},\dots ,a_{1,p})\sigma ,(a_{2,1},\dots ,a_{2,p})],\end{array}$$

where

$$\begin{array}{cc}\hfill a_{1,i}& =e,for1\le i\le p-1,\hfill \\ \hfill a_{2,1}& ={\left(f^{-1}\right)}^{r_1^{-1}\dots r_{p-1}^{-1}},\hfill \\ \hfill a_{2,i}& =r_{i-1}{a}_{2,i-1},for2\le i\le p,\hfill \\ \hfill a_{1,p}& =g^{a_{2,p}^{-1}}.\hfill \end{array}$$

Proof. 

Consider the following commutator

$$\begin{array}{cc}\hfill \kappa & =(a_{1,1},\dots ,a_{1,p})\sigma ·(a_{2,1},\dots ,a_{2,p})·(a_{1,p}^{-1},a_{1,1}^{-1},\dots ,a_{1,p-1}^{-1}){\sigma }^{-1}·(a_{2,1}^{-1},\dots ,a_{2,p}^{-1})\hfill \\ \hfill & =(a_{3,1},\dots ,a_{3,p}),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill a_{3,i}=a_{1,i}{a}_{2,1+(imodp)}{a}_{1,i}^{-1}{a}_{2,i}^{-1}.\end{array}$$

At first we compute the following

$$\begin{array}{c}\hfill a_{3,i}=a_{1,i}{a}_{2,i+1}{a}_{1,i}^{-1}{a}_{2,i}^{-1}=a_{2,i+1}{a}_{2,i}^{-1}=r_i{a}_{2,i}{a}_{2,i}^{-1}=r_i,for1\le i\le p-1.\end{array}$$

Then we make some transformation of $a_{3,p}$:

$$\begin{array}{cc}\hfill a_{3,p}& =a_{1,p}{a}_{2,1}{a}_{1,p}^{-1}{a}_{2,p}^{-1}\hfill \\ \hfill & =\left(a_{2,1}{a}_{2,1}^{-1}\right)a_{1,p}{a}_{2,1}{a}_{1,p}^{-1}{a}_{2,p}^{-1}\hfill \\ \hfill & =a_{2,1}[a_{2,1}^{-1},a_{1,p}]a_{2,p}^{-1}\hfill \\ \hfill & =a_{2,1}{a}_{2,p}^{-1}{a}_{2,p}[a_{2,1}^{-1},a_{1,p}]a_{2,p}^{-1}\hfill \\ \hfill & ={\left(a_{2,p}{a}_{2,1}^{-1}\right)}^{-1}[{\left(a_{2,1}^{-1}\right)}^{a_{2,p}},a_{1,p}^{a_{2,p}}]\hfill \\ \hfill & ={\left(a_{2,p}{a}_{2,1}^{-1}\right)}^{-1}[{\left(a_{2,1}^{-1}\right)}^{a_{2,p}{a}_{2,1}^{-1}},a_{1,p}^{a_{2,p}}].\hfill \end{array}$$

Now we can see that the form of the commutator $\kappa$ is similar to the form of w.

Introduce the following notation

$$\begin{array}{c}\hfill r^{\prime }=r_{p-1}\dots r_1.\end{array}$$

We note that from the definition of $a_{2,i}$ for $2\le i\le p$ it follows that

$$\begin{array}{c}\hfill r_i=a_{2,i+1}{a}_{2,i}^{-1},\mathrm{for}1\le i\le p-1.\end{array}$$

Therefore

$$\begin{array}{cc}\hfill r^{\prime }& =\left(a_{2,p}{a}_{2,p-1}^{-1}\right)\left(a_{2,p-1}{a}_{2,p-2}^{-1}\right)\dots \left(a_{2,3}{a}_{2,2}^{-1}\right)\left(a_{2,2}{a}_{2,1}^{-1}\right)\hfill \\ \hfill & =a_{2,p}{a}_{2,1}^{-1}.\hfill \end{array}$$

Then

$${\left(a_{2,p}{a}_{2,1}^{-1}\right)}^{-1}={\left(r^{\prime }\right)}^{-1}=r_1^{-1}\dots r_{p-1}^{-1}.$$

Now we compute the following

$$\begin{array}{cc}\hfill {\left(a_{2,1}^{-1}\right)}^{a_{2,p}{a}_{2,1}^{-1}}& ={\left({\left({\left(f^{-1}\right)}^{r_1^{-1}\dots r_{p-1}^{-1}}\right)}^{-1}\right)}^{r^{\prime }}={\left(f^{{\left(r^{\prime }\right)}^{-1}}\right)}^{r^{\prime }}=f,\hfill \\ \hfill a_{1,p}^{a_{2,p}}& ={\left(g^{a_{2,p}^{-1}}\right)}^{a_{2,p}}=g.\hfill \end{array}$$

Finally, we conclude that

$$\begin{array}{c}\hfill a_{3,p}=r_1^{-1}\dots r_{p-1}^{-1}[f,g].\end{array}$$

Thus, the commutator $\kappa$ has the same form as w. □

For future using we formulate previous Lemma for the case $p=2$.

Corollary 1.

For any group B, suppose $w\in {(B\wr C_2)}^{\prime }$ is defined by the following wreath recursion

$$\begin{array}{c}\hfill w=(r_1,r_1^{-1}[f,g]),\end{array}$$

where $r_1,f,g\in B$. Then we can represent w as commutator

$$\begin{array}{c}\hfill w=[(e,a_{1,2})\sigma ,(a_{2,1},a_{2,2})],\end{array}$$

where

$$\begin{array}{cc}\hfill a_{2,1}& ={\left(f^{-1}\right)}^{r_1^{-1}},\hfill \\ \hfill a_{2,2}& =r_1{a}_{2,1},\hfill \\ \hfill a_{1,2}& =g^{a_{2,2}^{-1}}.\hfill \end{array}$$

Lemma 4.

For any group B and integer $p\ge 2$ the inequality

$$\begin{array}{c}\hfill cw(B\wr C_p)\le max(1,cw\left(B\right))\end{array}$$

holds.

Proof. 

By Lemma 1, we can represent any $w\in {(B\wr C_p)}^{\prime }$ as the following wreath recursion

$$\begin{array}{cc}\hfill w& =(r_1,r_2,\dots ,r_{p-1},r_1^{-1}\dots ,r_{p-1}^{-1}\prod _{j=1}^k[f_j,g_j])\hfill \\ \hfill & =(r_1,r_2,\dots ,r_{p-1},r_1^{-1}\dots ,r_{p-1}^{-1}[f_1,g_1])·\prod _{j=2}^k[(e,\dots ,e,f_j),(e,\dots ,e,g_j)],\hfill \end{array}$$

where $r_1,\dots ,r_{p-1},f_j,g_j\in B$ and $k\le cw\left(B\right)$. Now by the Lemma 3 we can see that w can be represented as a product of $max(1,cw(B\left)\right)$ commutators. □

Corollary 2.

If $W=C_{p_k}\wr \dots \wr C_{p_1}$ then $cw\left(W\right)=1$ for $k\ge 2$.

Proof. 

If $B=C_{p_k}\wr C_{p_{k-1}}$, then take into consideration that $cw\left(B\right)>0$ (because $C_{p_k}\wr C_{p_{k-1}}$ is not commutative group). Lemma 4 implies that $cw(C_{p_k}\wr C_{p_{k-1}})=1$, and using the inequality $cw(C_{p_k}\wr C_{p_{k-1}}\wr C_{p_{k-2}})\le max(1,cw\left(B\right))$ from Lemma 4 we obtain $cw(C_{p_k}\wr C_{p_{k-1}}\wr C_{p_{k-2}})=1$. Similarly, if $W=C_{p_k}\wr \dots \wr C_{p_1}$ we use inductive assumption for $C_{p_k}\wr \dots \wr C_{p_2}$ the associativity of a permutational wreath product, the inequality of Lemma 4 and the equality $cw(C_{p_k}\wr \dots \wr C_{p_2})=1$ to conclude that $cw\left(W\right)=1$. □

We define our partially ordered set M and directed system of finite wreath products of cyclic groups as the set of all finite wreath products of cyclic groups. We make of use directed set $\mathbb{N}$.

$$H_k=\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}$$

(6)

Moreover, it has already been proved in Corollary 3 that each group of the form $\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}$ has a commutator width equal to 1; i.e., $cw\left(\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}\right)=1$. A partially ordered set of a subgroups is ordered by relation of inclusion group as a subgroup. Define the injective homomorphism $f_{k,k+1}$ from the $\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}$ into $\underset{i=1}{\stackrel{k+1}{\wr }}{\mathcal{C}}_{p_i}$ by mapping a generator of active group ${\mathcal{C}}_{p_i}$ of $H_k$ in a generator of active group ${\mathcal{C}}_{p_i}$ of $H_{k+1}$.

In more detail, the injective homomorphism $f_{k,k+1}$ is defined as $g\mapsto g(e,...,e)$, where a generator $g\in \underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}$, $g(e,...,e)\in \underset{i=1}{\stackrel{k+1}{\wr }}{\mathcal{C}}_{p_i}$.

We therefore obtain an injective homomorphism from $H_k$ onto the subgroup $\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}$ of $H_{k+1}$.

Corollary 3.

The direct limit ${\displaystyle \underrightarrow{lim}}\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}$ of the direct system $〈f_{k,j},\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}〉$ has commutator width 1.

Proof. 

We make the transition to the direct limit in the direct system $〈f_{k,j},\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}〉$ of injective mappings from chain $e\to ...\to \underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}\to \underset{i=1}{\stackrel{k+1}{\wr }}{\mathcal{C}}_{p_i}\to \underset{i=1}{\stackrel{k+2}{\wr }}{\mathcal{C}}_{p_i}\to ...$.

Since all mappings in chains are injective homomorphisms, they have a trivial kernel. Therefore, the transition to a direct limit boundary preserves the property $cw\left(H\right)=1$, because each group $H_k$ from the chain is endowed by $cw\left(H_k\right)=1$.

The direct limit of the direct system is denoted by ${\displaystyle \underrightarrow{lim}}\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}$ and is defined as disjoint union of the $H_k$’s modulo a certain equivalence relation:

$$\underrightarrow{lim}\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}={}^{{\displaystyle \coprod _k}\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}}/{}_{\sim }.$$

Since every element g of ${\displaystyle \underrightarrow{lim}}\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}$ coincides with a correspondent element from some $H_k$ of direct system, then by the injectivity of the mappings for g the property $cw\left(\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}\right)=1$ also holds. Thus, it holds for the whole ${\displaystyle \underrightarrow{lim}}\underset{i=1}{\stackrel{k}{\wr }}{\mathcal{C}}_{p_i}$. □

Corollary 4.

For prime p and $k\ge 2$ we have $cw\left(Sy{l}_p\left(S_{p^k}\right)\right)=1$. For prime $p>2$ and $k\ge 2$ we have $cw\left(Sy{l}_p\left(A_{p^k}\right)\right)=1$.

Proof. 

Since $Sy{l}_p\left(S_{p^k}\right)\simeq \stackrel{k}{\underset{\mathrm{i}=1}{\wr }}{C}_p$ (see [25,26]), we have $cw\left(Sy{l}_p\left(S_{p^k}\right)\right)=1$. It is well known that in a case $p>2$ where we have $Sy{l}_p{S}_{p^k}\simeq Sy{l}_p{A}_{p^k}$ (see [15,23]), so we obtain $cw\left(Sy{l}_p\left(A_{p^k}\right)\right)=1$. □

Proposition 1.

There is an inclusion $B_k^{\prime }<G_k$ holds.

Proof. 

We use induction on k. For $k=1$ we have $B_k^{\prime }=G_k=\left\{e\right\}$. Fix some $g=(g_1,g_2)\in B_k^{\prime }$. Then $g_1{g}_2\in B_{k-1}^{\prime }$ by Lemma 1. As $B_{k-1}^{\prime }<G_{k-1}$ by the induction hypothesis therefore $g_1{g}_2\in G_{k-1}$ and by definition of $G_k$ it follows that $g\in G_k$. □

Corollary 5.

The set $G_k$ is a subgroup in the group $B_k$.

Proof. 

According to the recursively definition of $G_k$ and $B_k$, where $G_k=\{(g_1,g_2)\pi \in B_k\mid g_1{g}_2\in G_{k-1}\}k>1$, i.e. $G_k$ is subset of $B_k$ with condition $g_1{g}_2\in G_{k-1}$. The result follows from the fact that $G_{k-1}$ is a subgroup of $G_k$. It is easy to check the closedness by multiplication elements of $G_k$ with condition $g_1{g}_2,h_1{h}_2\in G_{k-1}$ because $G_{k-1}$ is subgroup so $g_1{g}_2{h}_1{h}_2\in G_{k-1}$ too. The inverses can be verified easily. □

Lemma 5.

For any $k\ge 1$ we have $|G_k|=|B_k|/2$.

Proof. 

Induction on k. For $k=1$ we have $|G_1|=1=|B_1/2|$. Every element $g\in G_k$ can be uniquely written as the following wreath recursion

$$g=(g_1,g_2)\pi =(g_1,g_1^{-1}x)\pi$$

where $g_1\in B_{k-1}$, $x\in G_{k-1}$ and $\pi \in C_2$. Elements $g_1,x$ and $\pi$ are independent; therefore, $|G_k|=2|B_{k-1}|·|G_{k-1}|=2|B_{k-1}|·|B_{k-1}|/2=|B_k|/2$. □

Corollary 6.

The group $G_k$ is a normal subgroup in the group $B_k$; i.e., $G_k◃B_k$.

Proof. 

There exists normal embedding (normal injective monomorphism) $\phi :G_k\to B_k$ [27] such that $G_k◃B_k$. Indeed, according to Lemma index $\left|B_k:G_k\right|=2$, so it is a normal subgroup; that is, a quotient subgroup ${}^{B_k}/{}_{C_2}\simeq G_k$. □

Theorem 1.

For any $k\ge 1$ we have $G_k\simeq Sy{l}_2{A}_{2^k}$.

Proof. 

Group $C_2$ acts on the set $X=\{1,2\}$. Therefore, we can recursively define sets $X^k$ on which group $B_k$ acts $X^1=X,$$X^k=X^{k-1}\times X\mathrm{for}k>1$. At first we define $S_{2^k}=Sym\left(X^k\right)$ and $A_{2^k}=Alt\left(X^k\right)$ for all integers $k\ge 1$. Then $G_k<B_k<S_{2^k}$ and $A_{2^k}<S_{2^k}$.

We already know [15] that $B_k\simeq Sy{l}_2\left(S_{2^k}\right)$. Since $|A_{2^k}|=|S_{2^k}|/2$, $|Sy{l}_2{A}_{2^k}|=|Sy{l}_2{S}_{2^k}|/2=|B_k|/2$. By Lemma 3 it follows that $|Sy{l}_2{A}_{2^k}|=|G_k|$. Therefore, it remains to show that $G_k<Alt\left(X^k\right)$.

Let us fix some $g=(g_1,g_2){\sigma }^i$ where $g_1,g_2\in B_{k-1}$, $i\in \{0,1\}$ and $g_1{g}_2\in G_{k-1}$. Then we can represent g as follows

$$g=(g_1{g}_2,e)·(g_2^{-1},g_2)·(e,e,){\sigma }^i.$$

In order to prove this theorem it is enough to show that $(g_1{g}_2,e),(g_2^{-1},g_2),(e,e,)\sigma \in Alt\left(X^k\right)$.

Elements $(e,e,)\sigma$ just switch letters $x_1$ and $x_2$ for all $x\in X^k$. Therefore, $(e,e,)\sigma$ is product of $|X^{k-1}|=2^{k-1}$ transpositions, and therefore, $(e,e,)\sigma \in Alt\left(X^k\right)$.

Elements $g_2^{-1}$ and $g_2$ have the same cycle type. Therefore, elements $(g_2^{-1},e)$ and $(e,g_2)$ also have the same cycle type. Let us fix the following cycle decompositions

$$\begin{array}{c}\hfill (g_2^{-1},e)={\sigma }_1·\dots ·{\sigma }_n,\\ \hfill (e,g_2)={\pi }_1·\dots ·{\pi }_n.\end{array}$$

Note that element $(g_2^{-1},e)$ acts only on letters like $x_1$, and element $(e,g_2)$ acts only on letters like $x_2$. Therefore, we have the following cycle decomposition

$$\begin{array}{c}\hfill (g_2^{-1},g_2)={\sigma }_1·\dots ·{\sigma }_n·{\pi }_1·\dots ·{\pi }_n.\end{array}$$

So, element $(g_2^{-1},g_2)$ has even number of odd permutations and then $(g_2^{-1},g_2)\in Alt\left(X^k\right)$.

Note that $g_1{g}_2\in G_{k-1}$ and $G_{k-1}=Alt\left(X^{k-1}\right)$ by induction hypothesis. Therefore, $g_1{g}_2\in Alt\left(X^{k-1}\right)$. As elements $g_1{g}_2$ and $(g_1{g}_2,e)$ have the same cycle type, $(g_1{g}_2,e)\in Alt\left(X^k\right)$. □

As it was proven by the author in [15], the Sylow 2-subgroup has structure $B_{k-1}⋉W_{k-1}$, where the definition of $B_{k-1}$ is the same that which was given in [15].

Recall that it was denoted by $W_{k-1}$ the subgroup of $Aut{X}^{\left[k\right]}$ such that it had active states only on $X^{k-1}$ and a number of such states that was even; i.e., $W_{k-1}◃S{t}_{G_k}(k-1)$ [21]. It was proven that the size of $W_{k-1}$ is equal to $2^{2^{k-1}-1},k>1$ and its structure is ${\left(C_2\right)}^{2^{k-1}-1}$. The following structural theorem characterizing the group $G_k$ was proven by us [15].

Theorem 2.

A maximal 2-subgroup of $Aut{X}^{\left[k\right]}$ that acts by even permutations on $X^k$ has the structure of the semidirect product $G_k\simeq B_{k-1}⋉W_{k-1}$ and isomorphic to $Sy{l}_2{A}_{2^k}$.

Note that $W_{k-1}$ is subgroup of stabilizer of $X^{k-1}$, i.e., $W_{k-1}<S{t}_{Aut{X}^{\left[k\right]}}(k-1)◃Aut{X}^{\left[k\right]}$ and is normal to $W_{k-1}◃Aut{X}^{\left[k\right]}$, because conjugation keeps a cyclic structure of permutation, so even permutation maps are even. Therefore, such conjugation induce an automorphism of $W_{k-1}$ and $G_k\simeq B_{k-1}⋉W_{k-1}$.

Remark 1.

As a consequence, the structure founded by us in [15] is fully consistent with the recursive group representation (used in this paper) based on the concept of wreath recursion [10].

Theorem 3.

Elements of $B_k^{\prime }$ have the following form $B_k^{\prime }=\{[f,l]\mid f\in B_k,l\in G_k\}=\{[l,f]\mid f\in B_k,l\in G_k\}$.

Proof. 

It is enough to show either $B_k^{\prime }=\{[f,l]\mid f\in B_k,l\in G_k\}$ or $B_k^{\prime }=\{[l,f]\mid f\in B_k,l\in G_k\}$, because if $f=[g,h]$, then $f^{-1}=[h,g]$.

We prove the proposition by induction on k. For the case $k=1$ we have $B_1^{\prime }=\langle e\rangle$.

Consider case $k>1$. According to Lemma 2 and Corollary 1 every element $w\in B_k^{\prime }$ can be represented as

$$\begin{array}{c}\hfill w=(r_1,r_1^{-1}[f,g])\end{array}$$

for some $r_1,f\in B_{k-1}$ and $g\in G_{k-1}$ (by induction hypothesis). By the Corollary 1 we can represent w as commutator of

$$\begin{array}{c}\hfill (e,a_{1,2})\sigma \in B_k\mathrm{and}(a_{2,1},a_{2,2})\in B_k,\end{array}$$

where

$$\begin{array}{cc}\hfill a_{2,1}& ={\left(f^{-1}\right)}^{r_1^{-1}},\hfill \\ \hfill a_{2,2}& =r_1{a}_{2,1},\hfill \\ \hfill a_{1,2}& =g^{a_{2,2}^{-1}}.\hfill \end{array}$$

If $g\in G_{k-1}$, then by the definition of $G_k$ and Corollary 6 we obtain $(e,a_{1,2})\sigma \in G_k$. □

Remark 2.

Let us to note that Theorem 3 improve Corollary 4 for the case $Sy{l}_2{S}_{2^k}$.

Proposition 2.

If g is an element of the group $B_k$ then $g^2\in B_k^{\prime }$.

Proof. 

Induction on k. We note that $B_k=B_{k-1}\wr C_2$. Therefore, we fix some element

$$g=(g_1,g_2){\sigma }^i\in B_{k-1}\wr C_2,$$

where $g_1,g_2\in B_{k-1}$ and $i\in \{0,1\}$. Let us to consider $g^2$. Then, two cases are possible:

$$\begin{array}{c}\hfill g^2=(g_1^2,g_2^2)or{g}^2=(g_1{g}_2,g_2{g}_1).\end{array}$$

In the second case we consider a product of coordinates $g_1{g}_2·g_2{g}_1=g_1^2{g}_2^{2}x$. Since according to the induction hypothesis $g_i^2\in B_k^{\prime }$, $i\le 2$ then $g_1{g}_2·g_2{g}_1\in B_k^{\prime }$ also according to Lemma 1 $x\in B_k^{\prime }$. Therefore, a following inclusion holds $(g_1{g}_2,g_2{g}_1)=g^2\in B_k^{\prime }$. In first case the proof is even simpler because $g_1^2,g_2^2\in B^{\prime }$ by the induction hypothesis. □

Lemma 6.

If an element $g=(g_1,g_2)\in G_k^{\prime }$ then $g_1,g_2\in G_{k-1}$ and $g_1{g}_2\in B_{k-1}^{\prime }$.

Proof. 

As $B_k^{\prime }<G_k$, it is therefore enough to show that $g_1\in G_{k-1}$ and $g_1{g}_2\in B_{k-1}^{\prime }$. Let us fix some $g=(g_1,g_2)\in G_k^{\prime }<B_k^{\prime }$. Then, Lemma 1 implies that $g_1{g}_2\in B_{k-1}^{\prime }$.

In order to show that $g_1\in G_{k-1}$, we firstly consider just one commutator of arbitrary elements from $G_k$

$$\begin{array}{c}\hfill f=(f_1,f_2)\sigma ,h=(h_1,h_2)\pi \in G_k,\end{array}$$

where $f_1,f_2,h_1,h_2\in B_{k-1}$, $\sigma ,\pi \in C_2$. The definition of $G_k$ implies that $f_1{f}_2,h_1{h}_2\in G_{k-1}$.

If $g=(g_1,g_2)=[f,h]$, then

$$g_1=f_1{h}_i{f}_j^{-1}{h}_k^{-1}$$

for some $i,j,k\in \{1,2\}$. Then

$$\begin{array}{c}\hfill g_1=f_1{h}_i{f}_j{\left(f_j^{-1}\right)}^2{h}_k{\left(h_k^{-1}\right)}^2=\left(f_1{f}_j\right)\left(h_i{h}_k\right)x{\left(f_j^{-1}{h}_k^{-1}\right)}^2,\end{array}$$

where x is product of commutators of $f_i,h_j$ and $f_i,h_k$; hence, $x\in B_{k-1}^{\prime }$.

It is enough to consider the first product $f_1{f}_j$. If $j=1$, then $f_1^2\in B_{k-1}^{\prime }$ by Proposition 2 if $j=2$ then $f_1{f}_2\in G_{k-1}$ according to definition of $G_k$; the same is true for $h_i{h}_k$. Thus, for any $i,j,k$ it holds that $f_1{f}_j,h_i{h}_k\in G_{k-1}$. Besides that, a square ${\left(f_j^{-1}{h}_k^{-1}\right)}^2\in B_k^{\prime }$ according to Proposition 2. Therefore, $g_1\in G_{k-1}$ because of Propositions 1 and 2, the same is true for $g_2$.

Now it remains to consider the product of some $f=(f_1,f_2),h=(h_1,h_2)$, where $f_1,h_1\in G_{k-1}$, $f_1{h}_1\in G_{k-1}$ and $f_1{f}_2,h_1{h}_2\in B_{k-1}^{\prime }$

$$\begin{array}{c}\hfill fh=(f_1{h}_1,f_2{h}_2).\end{array}$$

Since $f_1{f}_2,h_1{h}_2\in B_{k-1}^{\prime }$ by imposed condition in this item and taking into account that $f_1{h}_1{f}_2{h}_2=f_1{f}_2{h}_1{h}_{2}x$ for some $x\in B_{k-1}^{\prime }$, then $f_1{h}_1{f}_2{h}_2\in B_{k-1}^{\prime }$ by Lemma 1. In other words, closedness by multiplication holds, and so according to Lemma 1, we have element of commutator $G_k^{\prime }$. □

In the following theorem we prove two facts at once.

Theorem 4.

The following statements are true.

1.

An element $g=(g_1,g_2)\in G_k^{\prime }$ iff $g_1,g_2\in G_{k-1}$ and $g_1{g}_2\in B_{k-1}^{\prime }$.

2.

Commutator subgroup $G_k^{\prime }$ coincides with set of all commutators for $k\ge 3$

$$G_k^{\prime }=\{[f_1,f_2]\mid f_1\in G_k,f_2\in G_k\}.$$

Proof. 

For the case $k=1$ we have $G_1^{\prime }=\langle e\rangle$. So, further we consider the case $k\ge 2$. If $k=2$ then we have $G_2\simeq V_4$, where $V_4$ is the Klein four group. But $cw\left(V_4\right)=0$.

Sufficiency of the first statement of this theorem follows from the Lemma 6. So, in order to prove the necessity of the both statements it is enough to show that element

$$\begin{array}{c}\hfill w=(r_1,r_1^{-1}x),\end{array}$$

where $r_1\in G_{k-1}$ and $x\in B_{k-1}^{\prime }$, can be represented as a commutator of elements from $G_k$. By Proposition 3 we have $x=[f,g]$ for some $f\in B_{k-1}$ and $g\in G_{k-1}$. Therefore,

$$\begin{array}{c}\hfill w=(r_1,r_1^{-1}[f,g]).\end{array}$$

By the Corollary 1 we can represent w as a commutator of

$$\begin{array}{c}\hfill (e,a_{1,2})\sigma \in B_k\mathrm{and}(a_{2,1},a_{2,2})\in B_k,\end{array}$$

where $a_{2,1}={\left(f^{-1}\right)}^{r_1^{-1}},a_{2,2}=r_1{a}_{2,1},a_{1,2}=g^{a_{2,2}^{-1}}.$ It only remains to show that $(e,a_{1,2})\sigma ,(a_{2,1},a_{2,2})\in G_k$. Note the following

$$\begin{array}{cc}\hfill a_{1,2}=g^{a_{2,2}^{-1}}& \in G_{k-1}\mathrm{by}\mathrm{Corollary}6.\hfill \\ \hfill a_{2,1}{a}_{2,2}=a_{2,1}{r}_1{a}_{2,1}=r_1[r_1,a_{2,1}]a_{2,1}^2& \in G_{k-1}\mathrm{by}\mathrm{Propositions}1\mathrm{and}2.\hfill \end{array}$$

So we have $(e,a_{1,2})\sigma \in G_k$ and $(a_{2,1},a_{2,2})\in G_k$ by the definition of $G_k$. □

Proposition 3.

For arbitrary $g\in G_k$ the inclusion $g^2\in G_k^{\prime }$ holds.

Proof. 

Induction on k: elements of $G_1^2$ have form ${\left(\sigma \right)}^2=e$, where $\sigma =(1,2)$, so the statement holds. In a general case, when $k>1$, the elements of $G_k$ have the form $g=(g_1,g_2){\sigma }^i,g_1,g_2\in B_{k-1},i\in \{0,1\}$. Then we have two possibilities: $g^2=(g_1^2,g_2^2)\mathrm{or}{g}^2=(g_1{g}_2,g_2{g}_1).$

Firstly we show that $g_1^2\in G_{k-1},g_2^2\in G_{k-1}.$ According to Proposition 2, we have $g_1^2,g_2^2\in B_{k-1}^{\prime }$ and according to Proposition 1, we have $B_{k-1}^{\prime }<G_{k-1}$. Then, using Theorem 4$g^2=(g_1^2,g_2^2)\in G_k$.

Consider the second case $g^2=(g_1{g}_2,g_2{g}_1)$. Since $g\in G_k$, then, according to the definition of $G_k$, we have that $g_1{g}_2\in G_{k-1}$. By Proposition 1, and the definition of $G_k$, we obtain

$$\begin{array}{c}{g}_2{g}_1=g_1{g}_2{g}_2^{-1}{g}_1^{-1}{g}_2{g}_1=g_1{g}_2[g_2^{-1},g_1^{-1}]\in G_{k-1},\\ g_1{g}_2·g_2{g}_1=g_1{g}_2^2{g}_1=g_1^2{g}_2^2[g_2^{-2},g_1^{-1}]\in B_{k-1}^{\prime }.\end{array}$$

Note that $g_1^2,g_2^2\in B_{k-1}^{\prime }$ according to Proposition 2, $g_1^2{g}_2^2[g_2^{-2},g_1^{-1}]\in B_{k-1}^{\prime }$. Since $g_1{g}_2·g_2{g}_1\in B_{k-1}^{\prime }$ and $g_1{g}_2,g_2{g}_1\in G_{k-1}$, then, according to Lemma 6, we obtain $g^2=(g_1{g}_2,g_2{g}_1)\in G_k^{\prime }$. □

Statement 1.

The commutator subgroup is a subgroup of$G_k^2$; i.e.,$G{{}^{\prime }}_k<G_k^2$.

Proof. 

Indeed, an arbitrary commutator presented as the product of squares. Let $a,b\in G$ and set that $x=a,y=a^{-1}ba,z=a^{-1}{b}^{-1}$. Then $x^2{y}^2{z}^2=a^2{\left(a^{-1}ba\right)}^2{\left(a^{-1}{b}^{-1}\right)}^2=ab{a}^{-1}{b}^{-1}.$ In more detail: $a^2{\left(a^{-1}ba\right)}^2{\left(a^{-1}{b}^{-1}\right)}^2=a^2{a}^{-1}ba{a}^{-1}ba{a}^{-1}{b}^{-1}{a}^{-1}{b}^{-1}=abb{b}^{-1}{a}^{-1}{b}^{-1}=\left[a,b\right]$. In such way we obtain all commutators and their products. Thus, we generate by squares the whole $G{{}^{\prime }}_k$. □

Corollary 7.

For the Syllow subgroup $\left(Sy{l}_2{A}_{2^k}\right)$ the following equalities $Sy{l}_2^{\prime }\left(A_{2^k}\right)={\left(Sy{l}_2\left(A_{2^k}\right)\right)}^2$, $\mathsf{\Phi }\left(Sy{l}_2{A}_{2^k}\right)=Sy{l}_2^{\prime }\left(A_{2^k}\right)$, which are characteristic properties of special p-groups [28], are true.

Proof. 

As is well known, for an arbitrary group (also by Statement 1) the following embedding $G^{\prime }◃G^2$ holds. In view of the above Proposition 3, a reverse embedding for $G_k$ is true. Thus, the group $Sy{l}_2{A}_{2^k}$ has some properties of special p-groups; that is, $P^{\prime }=\mathsf{\Phi }\left(P\right)$ [28] because $G_k^2=G_k^{\prime }$ and so $\mathsf{\Phi }\left(Sy{l}_2{A}_{2^k}\right)=Sy{l}_2^{\prime }\left(A_{2^k}\right)$. □

Corollary 8.

Commutator width of the group $Sy{l}_2{A}_{2^k}$ is equal to 1 for $k\ge 3$, also $cw\left(Sy{l}_2{A}_4\right)=0$.

It immediately follows from item 2 of Theorem 4 and the fact that $Sy{l}_2{A}_4\simeq V_4$.

4. Minimal Generating Set

For the construction of minimal generating set, we used the representation of elements of group $G_k$ by portraits of automorphisms at restricted binary tree $Aut{X}^k$. For convenience we will identify elements of $G_k$ with their faithful representations by portraits of automorphisms from $Aut{X}^{\left[k\right]}$.

We denote by ${A|}_l$, a set of all functions $a_l$, such that $[\epsilon ,\dots ,\epsilon ,a_l,\epsilon ,\dots ]\in {\left[A\right]}_l$. Recall that according to [29], l-coordinate subgroup $U<G$ is the following subgroup.

Definition 1.

For an arbitrarry $k\in \mathbb{N}$ we call a $k-$coordinatesubgroup $U<G$ a subgroup, which is determined by k-coordinate sets ${\left[U\right]}_l$, $l\in \mathbb{N}$, if this subgroup consists of all Kaloujnine’s tableaux $a\in I$ for which ${\left[a\right]}_l\in {\left[U\right]}_l$.

We denote by $G_k\left(l\right)$ a level subgroup of $G_k$, which consists of the tuples of v.p. from $X^l$, $l<k-1$ of any $\alpha \in G_k$. We denote as $G_k(k-1)$ such subgroups of $G_k$ that are generated by v.p., which are located on $X^{k-1}$ and isomorphic to $W_{k-1}$. Note that $G_k\left(l\right)$ is in bijective correspondence (and isomorphism) with l-coordinate subgroup ${\left[U\right]}_l$ [29].

For any v.p. $g_{li}$ in $v_{li}$ of $X^l$ we set in correspondence with $g_{li}$ the permutation $\phi \left(g_{li}\right)\in S_2$ by the following rule:

$$\phi \left(g_{li}\right)=\left\{\begin{array}{cc}\hfill (1,2),& \mathrm{if}{g}_{li}\ne e,\hfill \\ \hfill e,& \mathrm{if}{g}_{li}=e.\hfill \end{array}\right.$$

(7)

Define a homomorphic map from $G_k\left(l\right)$ onto $S_2$ with the kernel consisting of all products of even number of transpositions that belong to $G_k\left(l\right)$. For instance, the element $\left(12\right)\left(34\right)$ of $G_k\left(2\right)$ belongs to $ker\phi$. Hence, $\phi \left(g_{li}\right)\in S_2$.

Definition 2.

We define the subgroup of l-th level as a subgroup generated by all possible vertex permutation of this level.

Statement 2.

In ${G_k}^{\prime }$, the following k equalities are true:

$$\begin{array}{c}\prod _{l=1}^{2^l}\phi \left(g_{lj}\right)=e,0\le l<k-1.\end{array}$$

(8)

For the case $i=k-1$, the following condition holds:

$$\begin{array}{c}\prod _{j=1}^{2^{k-2}}\phi \left(g_{k-1j}\right)=\prod _{j=2^{k-2}+1}^{2^{k-1}}\phi \left(g_{k-1j}\right)=e.\end{array}$$

(9)

Thus, $G{{}^{\prime }}_k$ has k new conditions on a combination of level subgroup elements, except for the condition of last level parity from the original group.

Proof. 

Note that the condition (8) is compatible with those which were founded by R. Guralnik in [8], because as it was proven by author [15] $G_{k-1}\simeq B_{k-2}⋊{\mathcal{W}}_{k-1}$, where $B_{k-2}\simeq \underset{i=1}{\stackrel{k-2}{\wr }}{C}_2^{\left(i\right)}$.

According to Property 1, $G{{}^{\prime }}_k\le G_k^2$, so it is enough to prove the statement for the elements of $G_k^2$. Such elements, as it was described above, can be presented in the form $s=(s_{l1},...,s_{l{2}^l})\sigma$, where $\sigma \in G_{l-1}^{}$ and $s_{li}$ are states of $s\in G_k$ in $v_{li}$, $i\le 2^l$. For convenience we will make the transition from the tuple $(s_{l1},...,s_{l{2}^l})$ to the tuple $(g_{l1},...,g_{l{2}^l})$. Note that there is the trivial vertex permutation $g_{lj}^2=e$ in the product of the states $s_{lj}·s_{lj}$.

Since in $G{{}^{\prime }}_k$ v.p. on $X^0$ are trivial, so $\sigma$ can be decomposed as $\sigma =\left({\sigma }_{11},{\sigma }_{21}\right)$, where ${\sigma }_{21},{\sigma }_{22}$ are root permutations in $v_{11}$ and $v_{12}$.

Consider the square of s. We calculate squares ${\left(\left(s_{l1},s_{l2},...,s_{l{2}^{l-1}}\right)\sigma \right)}^2$. The condition (8) is equivalent to the condition that $s^2$ has even index on each level. Two cases are feasible: if permutation $\sigma =e$, then ${\left(\left(s_{l1},s_{l2},...,s_{l{2}^{l-1}}\right)\sigma \right)}^2=\left(s_{l1}^2,s_{l2}^2,...,s_{l{2}^{l-1}}^2\right)e$, so after the transition from $\left(s_{l1}^2,s_{l2}^2,...,s_{l{2}^{l-1}}^2\right)$ to $\left(g_{l1}^2,g_{l2}^2,...,g_{l{2}^{l-1}}^2\right)$, we get a tuple of trivial permutations $\left(e,...,e\right)$ on $X^l$, because $g_{lj}^2=e$. In the general case, if $\sigma \ne e$, after such transition we obtain $\left(g_{l1}{g}_{l\sigma \left(2\right)},...,g_{l{2}^{l-1}}{g}_{l\sigma \left(2^{l-1}\right)}\right){\sigma }_{}^2$. Consider the product of form

$$\begin{array}{c}\prod _{j=1}^{2^l}\phi \left(g_{lj}{g}_{l\sigma \left(j\right)}\right),\end{array}$$

(10)

where $\sigma$ and $g_{li}{g}_{l\sigma \left(i\right)}$ are from $\left(g_{l1}{g}_{l\sigma \left(2\right)},...,g_{l{2}^{l-1}}{g}_{l\sigma \left(2^{l-1}\right)}\right){\sigma }_{}^2$.

Note that each element $g_{lj}$ occurs twice in (10) regardless of the permutation $\sigma$; therefore, considering the commutativity of homomorphic images $\phi \left(g_{lj}\right),1\le j\le 2^l$ we conclude that ${\displaystyle \prod _{j=1}^{2^l}}\phi \left(g_{lj}{g}_{l\sigma \left(j\right)}\right)={\displaystyle \prod _{j=1}^{2^l}}\phi \left(g_{lj}^2\right)=e,$ because of $g_{lj}^2=e$. We rewrite ${\displaystyle \prod _{j=1}^{2^l}}\phi \left(g_{lj}^2\right)=e$ as characteristic condition: ${\displaystyle \prod _{j=1}^{2^{l-1}}}\phi \left(g_{lj}\right)={\displaystyle \prod _{j=2^{l-1}+1}^{2^l}}\phi \left(g_{lj}\right)=e.$

According to Property 1, any commutator from $G{{}^{\prime }}_k$ can be presented as a product of some squares $s^2,s\in G_k$, $s=\left((s_{l1},...,s_{l{2}^l})\sigma \right)$.

A product of elements of $G_k(k-1)$ satisfies the equation $\prod _{j=1}^{2^l}\phi \left(g_{lj}\right)=e$, because any permutation of elements from $X^k$, which belongs to $G_k$ is even. Consider the element $s=(s_{k-1,1},...,s_{k-1,2^{k-1}})\sigma$, where $(s_{k-1,1},...,s_{k-1,2^{k-1}})\in G_k(k-1)$, $\sigma \in G_{k-1}$. If $g_{01}=(1,2)$, where $g_{01}$ is root permutation of $\sigma$, then $s^2=(s_{k-1,1}{s}_{k-1\sigma \left(1\right)},...,s_{k-1,\left(2^{k-1}\right)}{s}_{k-1,\sigma \left(2^{k-1}\right)})$, where $\sigma \left(j\right)>2^{k-1}$ for $j\le 2^{k-1}$. And if $j<2^{k-1}$ then $\sigma \left(j\right)\ge 2^{k-1}$. Because of $\prod _{j=1}^{2^{k-1}}\phi \left(g_{k-1,j}\right)=e$ holds in $G_k$ and the property $\sigma \left(j\right)\le 2^{k-1}$ hold for $j>2^{k-1}$, then the product $\prod _{j=1}^{2^{k-2}}\phi (g_{k-1,j}{g}_{k-1,\sigma \left(j\right)})$ of images of v.p. from $(g_{k-1,1}{g}_{k-1,\sigma \left(1\right)},...,g_{k-1,\left(2^{k-1}\right)}{g}_{k-1,\sigma \left(2^{k-1}\right)})$ is equal to $\prod _{j=1}^{2^{k-1}}\phi (g_{k-1,j})=e$. Indeed, the products $\prod _{j=1}^{2^{k-1}}\phi \left(g_{k-1,j}\right)$ and $\prod _{j=1}^{2^{k-1}}\phi (g_{k-1,j}{g}_{k-1,\sigma \left(j\right)})$ have the same v.p. from $X^{k-1}$ which do not depend on such $\sigma$ as described above.

The same is true for right half of $X^{k-1}$. Therefore, the equality (9) holds.

Note that such product $\prod _{j=1}^{2^{k-1}}\phi \left(g_{k-1,j}\right)$ is homomorphic image of $(g_{l,1}{g}_{l,\sigma \left(1\right)},...,g_{l,\left(2^l\right)}{g}_{l\sigma \left(2^l\right)})$, where $l=k-1$, as an element of $G_k^{\prime }\left(l\right)$ after mapping (7).

If $g_{01}=e$, where $g_{01}$ is root permutation of $\sigma$, then $\sigma$ can be decomposed as $\sigma =\left({\sigma }_{11},{\sigma }_{12}\right)$, where ${\sigma }_{11},{\sigma }_{12}$ are root permutations in $v_{11}$ and $v_{12}$. As a result $s^2$ has a form $\left((s_{l1}{s}_{l\sigma \left(1\right)},...,s_{l\sigma \left(2^{l-1}\right)}){\sigma }_1^2,(s_{l{2}^{l-1}+1}{s}_{l\sigma (2^{l-1}+1)},...,s_{l\left(2^l\right)}{s}_{l\sigma \left(2^l\right)}){\sigma }_2^2\right)$, where $l=k-1$. As a result of action of ${\sigma }_{11}$ all states of l-th level with number $1\le j\le 2^{k-2}$ permutes in the set of coordinate from 1 to $2^{k-2}$. The others are fixed. The action of ${\sigma }_{11}$ is analogous.

It corresponds to the next form of element from $G_k^{\prime }\left(l\right)$: $(g_{l1}{g}_{l{\sigma }_1\left(1\right)},...,g_{l{\sigma }_1\left(2^{l-1}\right)}),(g_{l{2}^{l-1}+1}{g}_{l{\sigma }_2(2^{l-1}+1)},...,g_{l\left(2^l\right)}{g}_{l{\sigma }_2\left(2^l\right)})$. Therefore, the equality of form $\prod _{j=1}^{2^{k-2}}\phi \left(g_{k-1,j}{g}_{l\sigma \left(j\right)}\right)=\prod _{j=2^{k-2}+1}^{2^{k-1}}\phi \left(g_{k-1,j}^2\right)=e,$ because of $g_{k-1,j}^2=e$ holds. Thus, characteristic equation (9) of $k-1$ level holds.

The conditions (8) and (9) for every $s^2,s\in G_k$ hold, so they hold for their product that is equivalent to conditions which hold for every commutator. □

Definition 3.

We define a subdirect product of group $G_{k-1}$ with itself by equipping it with condition (8) and (9) of index parity on all of $k-1$ levels.

Corollary 9.

The subdirect product $G_{k-1}⊠G_{k-1}$ is defined by $k-2$ outer relations on level subgroups. The order of $G_{k-1}⊠G_{k-1}$ is $2^{2^k-k-2}$.

Proof. 

We specify a subdirect product for the group $G_{k-1}⊠G_{k-1}$ by using $(k-2)$ conditions for the subgroup levels. Each $G_{k-1}$ has even index on $k-2$-th level; it implies that its relation for $l=k-1$ holds automatically. This occurs because of the conditions of parity for the index of the last level is characteristic of each of the multipliers $G_{k-1}$. Therefore, It is not an essential condition for determining a subdirect product.

Thus, to specify a subdirect product in the group $G_{k-1}⊠G_{k-1}$, one need only $k-2$ outer conditions on subgroups of levels. Any of such conditions reduces the order of $G_{k-1}\times G_{k-1}$ by two times. Hence, taking into account that the order of $G_{k-1}$ is $2^{2^{k-1}-2}$, we can conclude that the order of $G_{k-1}⊠G_{k-1}$ as a subgroup of $G_{k-1}\times G_{k-1}$ is the following: $\left|G_{k-1}⊠G_{k-1}\right|={\left(2^{2^{k-1}-2}\right)}^2:2^{k-2}=2^{2^k-4}:2^{k-2}=2^{2^k-k-2}$. Thus, we use $k-2$ additional conditions on level subgroup to define the subdirect product $G_{k-1}⊠G_{k-1}$, which contain $G{{}^{\prime }}_k$ as a proper subgroup of $G_k$, because according to the conditions, which are realized in the commutator of $G{{}^{\prime }}_k$, (9) and (8) indexes of levels are even. □

Corollary 10.

A commutator $G{{}^{\prime }}_k$ is embedded as a normal subgroup in $G_{k-1}⊠G_{k-1}$.

Proof. 

A proof of injective embedding $G{{}^{\prime }}_k$ into $G_{k-1}⊠G_{k-1}$ immediately follows from last item of proof of Corollary 9. The minimality of $G{{}^{\prime }}_k$ as a normal subgroup of $G_k$ and injective embedding $G{{}^{\prime }}_k$ into $G_{k-1}⊠G_{k-1}$ immediately entails that $G{{}^{\prime }}_k◃G_{k-1}⊠G_{k-1}$. □

Theorem 5.

A commutator subgroup of $G_k$ has form $G{{}^{\prime }}_k=G_{k-1}⊠G_{k-1}$, where the subdirect product is defined by relations (8) and (9). The order of $G{{}^{\prime }}_k$ (the commutator subgroup of $Sy{l}_2{A}_{2^k}$) is $2^{2^k-k-2}$.

Proof. 

Since according to Statement 2 $(g_1,g_2)$ as elements of $G{{}^{\prime }}_k$ also satisfy relations (8) and (9), which define the subdirect product $G_{k-1}⊠G_{k-1}$.

Also $g_1{g}_2\in B{{}^{\prime }}_{k-1}$ implies the parity of permutation defined by $(g_1,g_2)$, because $B{{}^{\prime }}_{k-1}$ contains only an element with even index of level [15]. The group $G{{}^{\prime }}_k$ has two disjoint domains of transitivity so $G{{}^{\prime }}_k$ has the structure of a subdirect product of $G_{k-1}$ which acts on this domains transitively. Thus, all elements of $G{{}^{\prime }}_k$ satisfy the conditions (8) and (9) which define subdirect product $G_{k-1}⊠G_{k-1}$. Hence $G{{}^{\prime }}_k<G_{k-1}⊠G_{k-1}$ but $G{{}^{\prime }}_k$ can be equipped by some other relations; therefore, the presence of isomorphism has not yet been proven. For proving revers inclusion we have to show that every element from $G_{k-1}⊠G_{k-1}$ can be expressed as some word $a^{-1}{b}^{-1}ab$, where $a,b\in G_k$. Therefore, it suffices to show the reverse inclusion. For this goal we use the fact that $G{{}^{\prime }}_k<G_{k-1}⊠G_{k-1}$. Recall that is known [15] that the order of $G_k$ is $2^{2^k-2}$.

As it was shown above, $G{{}^{\prime }}_k$ has k new conditions relatively to $G_k$. Each condition is valid in some level-subgroup. Each of condition reduces an order of the corresponding level subgroup 2 times, so the order of $G{{}^{\prime }}_k$ is $2^k$ times smaller. On every $X^l$, $l\le k-1$, we have an even number of active v.p., by this reason there is the trivial permutation on $X^0$.

According to the Corollary 9, in the subdirect product $G_{k-1}⊠G_{k-1}$ there are exactly $k-2$ conditions relative to $G_{k-1}\times G_{k-1}$, which are for the subgroups of levels. It has been shown that the relations (8) and (9) are fulfilled in $G{{}^{\prime }}_k$.

Let ${\alpha }_{lm}$, $0\le l\le k-1$, $0\le m\le 2^{l-1}$ be an automorphism from $G_k$ having only one active v.p. in $v_{lm}$, and let ${\alpha }_{lm}$ have trivial permutations in rest of the vertices, so we can identify ${\alpha }_{lm}$ with a vertex permutation $g_{lm}$. Recall that partial case of notation of form ${\alpha }_{lm}$ is the generator ${\alpha }_l:={\alpha }_{l1}$ of $G_k$ which was defined by us in [15] and denoted by us as ${\alpha }_l$. Note that the order of ${\alpha }_{li},0\le l\le k-1$ is 2. Thus, ${\alpha }_{ji}={\alpha }_{ji}^{-1}$. We choose a generating set consisting of the following $2k-3$ elements: $\left({\alpha }_{1,1;2}\right),{\alpha }_{2,1},...,{\alpha }_{k-1,1},{\alpha }_{2,3},...,{\alpha }_{k-{1,2}^{k-2}+1}$, where $\left({\alpha }_{1,1;2}\right)$ is an automorphism having exactly two active v.p.s in $v_{11}$ and $v_{12}$. Products of the form $\left({\alpha }_{j1}{\alpha }_{l1}{\alpha }_{j1}\right){\alpha }_{l1}$ are denoted by $P_{lj}$. Using a conjugation by generator ${\alpha }_j$, $0\le j<l$ we can express any v.p. on l-level, because $\left({\alpha }_j{\alpha }_l{\alpha }_j\right)={\alpha }_{l{2}^{l-j-1}+1}$. Defime the product $P_{lj}=\left({\alpha }_j{\alpha }_l{\alpha }_j\right){\alpha }_l$. Consider an algorithm of constructing any element of $G_{k-1}⊠G_{k-1}$ as a product of commutators.

• We need to show that every element of $G_{k-1}⊠G_{k-1}$ satisfying the relations (8), (9) can be constructed as ${\alpha }^{-1}{\beta }^{-1}\alpha \beta$, $\alpha ,\beta \in G_k$.
  This proves the absence of other relations in $G{{}^{\prime }}_k$ except those that in the subdirect product $G_{k-1}⊠G_{k-1}$. Thereby we prove the embeddedness of $G{{}^{\prime }}_k$ in $G_{k-1}⊠G_{k-1}$. We have to construct an element of form $P_{k-1}{P}_{k-2}·...·P_1{P}_0$ as a product of elements of form $P_l=\prod _{t=1}^{m_l}{P}_{l{j}_t}$ satisfying relations (8) and (9). Where $P_{lj}=\left({\alpha }_j{\alpha }_l{\alpha }_j\right){\alpha }_l$ is commutator of ${\alpha }_l,{\alpha }_j$.
• We have to construct an automorphism which has an arbitrary tuple of two active v.p.s satisfying the relations (8) and (9) on $X^l$ as a product of $P_{lj}$ and $P_{li}$. We use the generator ${\alpha }_l$ and conjugate by ${\alpha }_j$, $j<l$. This corresponds to the tuple of v.p. of the form $(g_{l1},e,...,e,g_{lj},e,...,e)$, where $g_{l1},g_{lj}$ are non-trivial. Note that this tuple $(g_{l1},e,...,e,g_{li},e,...,e)$, which corresponds to $P_{li}$, is an element of direct product if we consider ${\alpha }_{lj}$ as an element of $S_2$ in vertices of $X^l$. To obtain a tuple of v.p. of form $(e,...,e,g_{li},e,...,e,g_{lj},e,...,e)\in G_k\left(l\right)$ we simply multiply $P_{lj}$ and $P_{li}\in G_k\left(l\right)$.
• To obtain a tuple T of v.p. with $2m$ active v.p. satisfying the relations (8), (9) we construct $P_l=\prod _{t=1}^{m_l}{P}_{l{j}_t},m<2^l$ for varying $j_t\le 2^l$, where the values of $j_t$ correspond to the second coordinate of active v.p. from the tuple T, which we have to construct. To construct an arbitrary element h we form a corresponding product $h=\prod _{l=1}^k{P}_l$. On the $(k-1)$-th level, we choose the generator $\tau$ to be $\tau ={\tau }_{k-1,1}{\tau }_{k-1,2^{k-1}}$, as defined in [15].

Since h satisfies the relations (8) and (9) for all $0\le l\le k$ then $h\in G_{k-1}⊠G_{k-1}$.

On the $(k-1)$-th level, we choose the generator $\tau$ which was defined in [15] as $\tau ={\tau }_{k-1,1}{\tau }_{k-1,2^{k-1}}$. Recall that it was shown in [15] how to express any ${\tau }_{ij}$ using $\tau$, ${\tau }_{i,2^{k-2}}$, ${\tau }_{j,2^{k-2}}$, where $i,j<2^{k-2}$, in form of a product of commutators ${\tau }_{ij}={\tau }_{i,2^{k-2}}{\tau }_{j,2^{k-2}}=\left({\alpha }_i^{-1}{\tau }_{{1,2}^{k-2}}^{-1}{\alpha }_i^{}{\tau }_{{j,2}^{k-2}}^{}\right)$.

Here ${\tau }_{i{,2}^{k-2}}$ was expressed as the commutator ${\tau }_{i{,2}^{k-2}}={\alpha }_i^{-1}{\tau }_{{1,2}^{k-2}}^{-1}{\alpha }_i^{}{\tau }_{{1,2}^{k-2}}^{}$.

Thus, we express all tuples of elements satisfying to relations (8) and (9) by using only commutators of $G_k$. Thus, we get all tuples of each level subgroup elements satisfying the relations (8) and (9). This means we express every element of each level subgroup by commutators. In particular, to obtain a tuple of v.p. with $2m$ active v.p. on $X^{k-2}$ of $v_{11}{X}^{[k-1]}$, we will construct the product for ${\tau }_{ij}$ for varying $i,j<2^{k-2}$.

Thus, all vertex labelings of automorphisms, which appear in the representation of $G_{k-1}⊠G_{k-1}$ by portraits as the subgroup of $Aut{X}^{\left[k\right]}$, are also in the representation of $G{{}^{\prime }}_k$.

Since there are faithful representations of $G_{k-1}⊠G_{k-1}$ and ${G^{\prime }}_k$ by portraits of automorphisms from $Aut{X}^{\left[k\right]}$, which coincide with each other, subgroup ${G^{\prime }}_k$ of $G_{k-1}⊠G_{k-1}\simeq G{{}^{\prime }}_k$ is equal to $G_{k-1}⊠G_{k-1}$ (i.e., $G_{k-1}⊠G_{k-1}=G{{}^{\prime }}_k$). □

The archived results are confirmed by algebraic system GAP calculations. For instance, $\left|Sy{l}_2{A}_8\right|=2^6=2^{2^3-2}$ and $\left|{\left(Syl{A}_{2^3}\right)}^{\prime }\right|=2^{2^3-3-2}=8$. The order of $G_2$ is 4, the number of additional relations in the subdirect product is $k-2=3-2=1$. We have the same result $(4·4):2^1=8$, which confirms Theorem 5.

Example 1.

Set $k=4$ then $|{\left(Syl{A}_{16}\right)}^{\prime }|=|{\left(G_4\right)}^{\prime }|=1024$, $|G_3|=64$, since $k-2=2$, so according to our theorem above order of $Sy{l}_2{A}_{16}⊠Sy{l}_2{A}_{16}$ is defined by $2^{k-2}=2^2$ relations, and by this reason is equal to $(64·64):4=1024$. Thus, orders are coincides.

Example 2.

The true order of ${\left(Sy{l}_2{A}_{32}\right)}^{\prime }$ is $33554432=2^{25}$, $k=5$. A number of additional relations which define the subdirect product is $k-2=3$. Thus, according to Theorem 5, $\mid {(Sy{l}_2{A}_{16}⊠Sy{l}_2{A}_{16})}^{\prime }\mid =2^{14}{2}^{14}:2^{5-2}=2^{28}:2^{5-2}=2^{25}$.

According to calculations in GAP we have: $Sy{l}_2{A}_7\simeq Sy{l}_2{A}_6\simeq D_4$. Therefore, its derived subgroup ${\left(Sy{l}_2{A}_7\right)}^{\prime }\simeq {\left(Sy{l}_2{A}_6\right)}^{\prime }\simeq {\left(D_4\right)}^{\prime }=C_2$.

The following structural law for Syllows 2-subgroups is typical. The structures of $Sy{l}_2{A}_n$ and $Sy{l}_2{A}_k$ are the same if all n and k have the same multiple of two as the multiplier in decomposition on $n!$ and $k!$ Thus, $Sy{l}_2{A}_{2k}\simeq Sy{l}_2{A}_{2k+1}$.

Example 3.

$Sy{l}_2{A}_7\simeq Sy{l}_2{A}_6\simeq D_4$, $Sy{l}_2{A}_{10}\simeq Sy{l}_2{A}_{11}\simeq Sy{l}_2{S}_8\simeq \left(D_4\times D_4\right)⋊C_2$. $Sy{l}_2{A}_{12}\simeq Sy{l}_2{S}_8⊠Sy{l}_2{S}_4$, by the same reasons that from the proof of Corollary 9 its commutator subgroup is decomposed as ${\left(Sy{l}_2{A}_{12}\right)}^{\prime }\simeq {\left(Sy{l}_2{S}_8\right)}^{\prime }\times {\left(Sy{l}_2{S}_4\right)}^{\prime }$.

Lemma 7.

In $G_k^{\prime \prime }$ the following equalities are true:

$$\begin{array}{c}\prod _{j=1}^{2^{l-2}}\phi \left(g_{lj}\right)=\prod _{j=2^{l-2}+1}^{2^{l-1}}\phi \left(g_{lj}\right)=\prod _{j=2^{l-1}+1}^{2^{l-1}+2^{l-2}}\phi \left(g_{lj}\right)=\prod _{j=2^{l-1}+2^{l-2}+1}^{2^l}\phi (g_{lj}),2<l<k.\end{array}$$

(11)

In case $l=k-1$, the following conditions hold:

$$\begin{array}{c}\prod _{j=1}^{2^{l-2}}\phi \left(g_{lj}\right)=\prod _{j=2^{i-1}+1}^{2^{l-1}}\phi \left(g_{lj}\right)=e,\prod _{j=2^{l-1}}^{2^{l-1}+2^{l-2}}\phi \left(g_{lj}\right)=\prod _{j=2^{l-1}+2^{l-2}}^{2^l}\phi \left(g_{lj}\right)=e.\end{array}$$

(12)

In other terms, the subgroup $G_k^{\prime \prime }$ has an even index of any level of $v_{11}{X}^{\left[k-2\right]}$ and of $v_{12}{X}^{\left[k-2\right]}$. The order of $G_k^{\prime \prime }$ is equal to $2^{2^k-3k+1}$.

Proof. 

As a result of derivation of $G_k^{\prime }$, elements of $G_k^{\prime \prime }\left(1\right)$ are trivial. Due the fact that $G{{}^{\prime }}_k\simeq G_{k-1}⊠G_{k-1}$, we can derivate $G{{}^{\prime }}_k$ by commponents. The commutator of $G_{k-1}$ is already investigated in Theorem 5. As $G_{k-1}^2=G{{}^{\prime }}_{k-1}$ by Corollary 7, it is more convenient to present a characteristic equalities in the second commutator $G^{\prime }{{}^{\prime }}_k\simeq G{{}^{\prime }}_{k-1}⊠G{{}^{\prime }}_{k-1}$ as equations in $G_{k-1}^2⊠G_{k-1}^2$. As shown above, for $2\le l<k-1$, in $G_{k-1}^2$ the following equalities are true:

$$\begin{array}{c}\prod _{j=1}^{2^{l-1}}\phi \left(g_{lj}{g}_{l\sigma \left(j\right)}\right)=\prod _{j=1}^{2^{l-1}}\phi \left(g_{lj}\right)\prod _{j=1}^{2^{l-1}}\phi \left(g_{l\sigma \left(j\right)}\right)=\prod _{j=1}^{2^{l-1}}\phi \left(g_{lj}\right)\prod _{j=1}^{2^{l-1}}\phi \left(g_{li}\right)=\prod _{j=1}^{2^{l-1}}\phi \left(g_{lj}^2\right)=e\end{array}$$

(13)

$$\begin{array}{c}\prod _{j=1}^{2^{l-2}}\phi \left(g_{lj}\right)=\prod _{j=2^{l-2}+1}^{2^{l-1}}\phi (g_{lj})=\prod _{j=2^{l-1}+1}^{2^{l-1}+2^{l-2}}\phi \left(g_{lj}\right)=\prod _{j=2^{l-1}+2^{l-2}+1}^{2^l}\phi (g_{lj}).\end{array}$$

(14)

The equality (14) holds since it is valid in the initial group $G{{}^{\prime }}_k\simeq G_{k-1}⊠G_{k-1}$. The equalities

$$\prod _{j=2^{l-1}+1}^{2^{l-1}+2^{l-2}}\phi \left(g_{lj}\right)=\prod _{j=2^{l-1}+2^{l-2}+1}^{2^l}\phi (g_{lj})$$

hold for elements of second group $G{{}^{\prime }}_{k-1}$, since the elements of the original group are endowed with these conditions.

In ${\left(G{{}^{\prime }}_k\right)}^2$ any element g of $G{{}^{\prime }}_k\left(l\right)$ satisfies the equality (14). Moreover, g satisfies the previous conditions (11) because of ${\left(G{}_{k-1}\left(l\right)\right)}^2=G{{}^{\prime }}_{k-1}\left(l\right)$.

The similar conditions appear in ${\left(G{{}^{\prime }}_{k-1}(k-2)\right)}^2$ after squaring of $G{{}^{\prime }}_k$. Thus, taking into account the characteristic equations of $G{{}^{\prime }}_{k-1}\left(l\right)$, the subgroup ${\left(G{{}^{\prime }}_{k-1}(k-2)\right)}^2$ satisfies the equality:

$$\begin{array}{c}\prod _{j=1}^{2^{k-3}}\phi \left(g_{lj}\right)=\prod _{j=2^{k-3}+1}^{2^{k-2}}\phi \left(g_{lj}\right)=e,\prod _{j=2^{k-2}+1}^{2^{k-2}+2^{k-3}}\phi \left(g_{lj}\right)=\prod _{j=2^{k-1}+2^{k-2}+1}^{2^{k-1}}\phi \left(g_{lj}\right)=e.\end{array}$$

(15)

Taking into account the structure $G{{}^{\prime }}_k\simeq G_{k-1}⊠G_{k-1}$, we obtain after the derivation $G^{\prime }{{}^{\prime }}_k\simeq (G_{k-2}⊠G_{k-2})⊠(G_{k-2}⊠G_{k-2})$. With respect to conditions (8) and (9) in the subdirect product, we have that the order of $G^{\prime }{{}^{\prime }}_k$ is $2^{2^k-k-2}:2^{2k-3}=2^{2^k-3k+1}$ because on each level $2\le l<k$, the order of level subgroup $G^{\prime }{{}^{\prime }}_k\left(l\right)$ is 4 times smaller than the order of $G{{}^{\prime }}_k\left(l\right)$. On the first level, one new condition arises that reduces the order of $G{{}^{\prime }}_k\left(1\right)$ by 2 times. In total, we have $2(k-2)+1=2k-3$ new conditions for comparing with $G{{}^{\prime }}_k$. □

Corollary 11.

Any minimal generating set of $Syl{{}^{\prime }}_2{A}_{2^k}$, $k>2$ consists of $2k-3$ elements.

Proof. 

The proof is based on two facts about $G_k^{{}^{\prime }2}{G}_k^{{}^{\prime \prime }}\simeq G_k^{{}^{\prime }2}={G^{\prime }}_k$. More precisely it is based on Corollary 7 and on a calculating of the index $\left|G^{\prime }:G_k^{{}^{\prime }2}{G}_k^{{}^{\prime \prime }}\right|=2^{2k-3}$.

To justify that the index $\left|G^{\prime }:G_k^{{}^{\prime }2}{G}_k^{{}^{\prime \prime }}\right|=2^{2k-3}$, we take into consideration the orders of these subgroups from Theorem 5 and Lemma 7. Corollary 7 tell us that the subgroup $G_k^2$ is equal to the subgroup $G_k^{\prime }$, then the Frattiny subgroup $\mathsf{\Phi }\left(G_k^{\prime }\right)=G_k^{\prime \prime }={G_k^{\prime }}^2$. According to Corollary 7 the subgroup $G_k^2$ is equal to the subgroup $G_k^{\prime }$, then the Frattiny subgroup $\mathsf{\Phi }\left(G_k^{\prime }\right)=G_k^{\prime \prime }={G_k^{\prime }}^2$. Further, for finding the Frattiny factor, which is an elementary abelian 2-group, it is enough sufficient to calculate $\left|G^{\prime }:G_k^{{}^{\prime \prime }}\right|$ because of $\mathsf{\Phi }\left(Sy{l}_2^{\prime }{A}_{2^k}\right)=Sy{l}_2^{\prime \prime }\left(A_{2^k}\right)$. Due to Lemma 7, we have $G^{\prime }{{}^{\prime }}_{k-1}\simeq G{{}^{\prime }}_{k-2}⊠G{{}^{\prime }}_{k-2}$, hence the order of $G^{\prime }{{}^{\prime }}_{k-1}$ is equal to $2^{2^k-3k+1}$. Taking into account that $G^{\prime }{{}^{\prime }}_k$ is normal subgroup of $G{{}^{\prime }}_k$, we compute the order of Frattiny quotient is $2^{2k-3}$. Thus, according to Frattiny theorem, a minimal generating set of $Syl{{}^{\prime }}_2{A}_{2^k}$ consists of $2k-3$ elements. It is well known [28], the orders of irreducible generating sets for p-group are equal to each other. □

In case $k=2$ the $Syl{}_2{A}_4\simeq K_4$, therefore the commutator subgroup is trivial.

Example 4.

The size of $\left(G_4^{\prime \prime }\right)$ is 32. The size of the direct product ${\left(G_3^{\prime }\right)}^2$ is 64, but, due to relation on second level of $G_k^{\prime \prime }$, the direct product ${\left(G_3^{\prime }\right)}^2$ transforms into the subdirect product $G_3^{\prime }⊠G_3^{\prime }$ that has two times less feasible combination on $X^2$. The number of additional relations in the subdirect product is $k-3=4-3=1$. Thus, the order of product is reduced by $2^1$ times.

Example 5.

The commutator subgroup of $Sy{l}_2^{\prime }\left(A_8\right)$ consists of elements: $\{e,(13\left)\right(24\left)\right(57\left)\right(68),(12\left)\right(34),(14\left)\right(23\left)\right(57\left)\right(68),(56\left)\right(78),(13\left)\right(24\left)\right(58\left)\right(67),(12\left)\right(34\left)\right(56\left)\right(78),(14\left)\right(23\left)\right(58\left)\right(67\left)\right\}$. The commutator $Sy{l}_2^{\prime }\left(A_8\right)\simeq C_2^3$ is an elementary abelian 2-group of order 8. This fact confirms our formula $d\left(G_k\right)=2k-3$, because $k=3$ and $d\left(G_k\right)=2k-3=3$. A minimal generating set of $Sy{l}_2^{\prime }\left(A_8\right)$ consists of three generators: $(1,3)(2,4)(5,7)(6,8),(1,2)(3,4),(1,3)(2,4)(5,8)(6,7).$

Example 6.

The minimal generating set of $Sy{l}_2^{\prime }\left(A_{16}\right)$ consists of five (that is $2·4-3$) generators: $(1,4,2,3)(5,6)(9,12)(10,11),(1,4)(2,3)(5,8)(6,7),(1,2)(5,6),(1,7,3,5)(2,8,4,6)(9,14,12,16)\times \times (10,13,11,15),(1,7)(2,8)(3,6)(4,5)(9,16,10,15)(11,14,12,13)$.

Example 7.

A minimal generating set of $Sy{l}_2^{\prime }\left(A_{32}\right)$ consists of seven (that is $2·5-3$) generators: $(23,24)(31,32),(1,7)(2,8)(3,5,4,6)(11,12)(25,32)(26,31)(27,29)(28,30),(3,4)(5,8)(6,7)(13,14)(23,24)(27,28)(29,32)(30,31),(7,8)(15,16)(23,24)(31,32),(1,9,7,15)(2,10,8,16)(3,11,5,13)(4,12,6,14)(17,29,22,27,18,30,21,28)(19,32,23,26,20,31,24,25),(1,5,2,6)(3,7,4,8)(9,15)(10,16)(11,13)(12,14)(19,20)(21,24,22,23)(29,31)(30,32),(3,4)(5,8)(6,7)(9,11,10,12)(13,14)(15,16)(17,23,20,22,18,24,19,21)(25,29,27,32,26,30,28,31).$

This confirms our formula of minimal generating set size $2·k-3$.

The minimal generating set for $G_4$ can be presented in form of wreath recursion:

$$a_1=(e,e)\sigma ,b_2=\left(a_1,e\right),a_3=\left(b_2,e\right),b_4=\left(b_3,b_3\right),$$

where $\sigma =(1,2)$. The minimal generating set for $G{{}^{\prime }}_4$ can be presented in form of wreath recursion:

$$a_2=(\sigma ,\sigma ),a_3=(e,a_2),a_4=\left(a_3,a_3\right),b_3=(e,b_2),b_4=(b_3,b_3).$$

where $\sigma ,a_3,a_4$ are generators of the first multiplier $G_3$ and $\sigma ,b_3,b_4$ are generators of the second.

5. Conclusions

The size of minimal generating set for commutator of Sylow 2-subgroup of alternating group $A_{2^k}$ was proven to be equal to $2k-3$, where $k>2$.

A new approach to presentation of Sylow 2-subgroups of alternating group $A_{2^k}$ was applied. As a result, the short proof of a fact that commutator width of Sylow 2-subgroups of the alternating group $A_{2^k}$ ($k>2$), permutation group $S_{2^k}$ and Sylow p-subgroups of $Sy{l}_2{A}_{p^k}$ ($Sy{l}_2{S}_{p^k}$) are equal to 1 was obtained. Commutator widths of permutational wreath products $B\wr C_n$ were investigated.

We constructed the minimal generating set of the commutator subgroup of the Sylow 2-subgroup of the alternating group. The inclusion problem [18] for $Sy{l}_2{A}_{2^k}$ and its subgroups as ${\left(Sy{l}_2{A}_{2^k}\right)}^{\prime }$ and ${\left(Sy{l}_2{A}_{2^k}\right)}^{\prime \prime }$ was investigated by us. The relation between solving of the inclusion problem of and conjugacy search problem [19] in this group was established by us.