Article The Derived Subgroups of Sylow 2-Subgroups of the Alternating Group, Commutator Width of Wreath Product of Groups
Ruslan V. Skuratovskii 1,2 1 Igor Sikorsky Kyiv Polytechnic Institute, National Technical University of Ukraine, Peremogy 37, 03056 Kiev, Ukraine; [email protected] or [email protected] 2 Interregional Academy of Personnel Management, 03039 Kiev, Ukraine
Received: 14 December 2019; Accepted: 31 January 2020; Published: 30 March 2020
Abstract: The structure of the commutator subgroup of Sylow 2-subgroups of an alternating group
A2k 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 A2k is 2 constructed. It is shown that (Syl2 A2k ) = Syl20 A2k , 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 Cp , pi N equals i ∈ to 1. The commutator width of direct limit of wreath product of cyclic groups is found. Upper bounds for the commutator width (cw(G)) of a wreath product of groups are presented in this paper.
A presentation in form of wreath recursion of Sylow 2-subgroups Syl2(A2k ) of A2k is introduced. As a result, a short proof that the commutator width is equal to 1 for Sylow 2-subgroups of alternating
group A2k , where k > 2, the permutation group S2k , as well as Sylow p-subgroups of Syl2 Apk as well as Syl S k ) are equal to 1 was obtained. A commutator width of permutational wreath product B C 2 p o n is investigated. An upper bound of the commutator width of permutational wreath product B C o 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.
Keywords: commutator subgroup; alternating group; minimal generating set; Sylow 2-subgroups; Sylow p-subgroups; commutator width; permutational wreath product
MSC: 20B05; 20D20; 20B25; 20B22; 20B07; 20E08; 20E28; 20B35; 20D10; 20B27
1. Introduction
The object of our study is the commutatorwidth [1] of Sylow 2-subgroups of alternating group A2k . 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 [G, G] there exist elements ∈ x1,..., xn, y1,..., yn in G such that g = [x1, y1] ... [xn, yn]. 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 n if and ≤ only if the equation [X1, X2] ... [X2n 1, X2n]g = 1 is solvable for all g G0. − ∈
Mathematics 2020, 8, 472; doi:10.3390/math8040472 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 472 2 of 19
As it is well known, the first example of a group G with commutator width greater than 1 (cw(G) > 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 B, where C is cyclic n o n group of order n, in terms of the commutator width cw(B) of passive group B. A form of commutators of wreath product A B was briefly considered in [9]. The form of commutator presentation [9] is o 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. We deduce an estimation for commutator width of wreath product C B, where C is cyclic In this paper we continue investigations started in [11–17]. We findn a≀ minimal generatingn set and group of order n, taking into consideration the cw(B) of passive group B. A form of commutators theof wreath structure product for commutatorA B was shortly subgroup considered of Syl2 inA [10].2k . The form of commutator presentation [10] ≀ is proposedResearch by of us commutator-group as wreath recursion serves [8] and the commutator decision of width inclusion of it was problem studied. [18 We] for impose elements of Sylweaker2 A2k conditionin its derived on the subgroup presentation(Syl2 ofA2 wreathk )0. Knowledge product commutator of the method then for it solving was prosed the ofby inclusion J. problemMeldrum. in a subgroup H facilitates the solution of the problem of finding the conjugate elements in theIn whole this papergroup we (conjugacy continue asearch investigations problem) stared [19]. Because in [19,20]. by We the find characterization a minimal generating of the conjugated set and the structure for commutator subgroup of Syl A k . elements g and h 1gh, we can determine which subgroups2 2 they belong to and which do not belong. A research of− commutator-group serve to decision of inclusion problem [9] for elements of It is known that the commutator width of iterated wreath products of nonabelian finite simple Syl2A2k in its derived subgroup (Syl2A2k )′. It was known that, the commutator width of iterated groupswreath is products bounded of by nonabelian an absolute finite constant simple [ groups7,20]. But is bounded it has not by been an absolute proven thatconstant commutator [3,13]. But subgroup k k ofit was notpi consists proven of that commutators. commutator subgroup We generalize of thepi consists passive of group commutators. of this wreath We generalize product to any i=o1 C i=1≀ C groupthe passiveB instead group of of only this wreath wreath product product toof anycyclic group groupsB instead and obtain of only an wreath exact commutator product of cyclic width. groupsAdditionally, and obtain an we exact are going commutator to prove width. that the commutator width of Sylow p-subgroups of symmetricAlso we and are alternating going to prove groups that for thep commutator2 is 1. width of Sylow p-subgroups of symmetric and alternating groups for p 2 is 1. ≥ ≥ 2. Preliminaries 2. Preliminaries Let GLetbeG abe group a group acting acting (from (from the right) the right) by permutations by permutations on a set onX aand set X letandH be let anH arbitrarybe an arbitrary X group.group. Then Then the the (permutational) (permutational) wreath wreath product productHH G isis the the semidirect semidirect product HX hGG,, where G acts X X o≀ onG theacts direct on the power directH powerbyH the respectiveby the respective permutations permutations of the of direct the direct factors. factors. The cyclic The cyclic group Cp or group Cp or (Cp,X) is equipped with a natural action by the left shift on X = 1, . . . , p , p N. (Cp, X) is equipped with a natural action by the left shift on X = 1, ..., p , p{ N. It} is well∈ known It is well known that a wreath product of permutation groups is associative{ } construction∈ [10]. that a wreath product of permutation groups is associative construction [9]. The multiplication rule of automorphisms g, h which presented in form of the wreath recursion The multiplication rule of automorphisms g and h, which are presented in form of the wreath [12] g = (g(1), g(2), . . . , g(d))σg, h = (h(1), h(2), . . . , h(d))σh, is given by the formula: recursion [21] g = (g(1), g(2),..., g(d))σg, h = (h(1), h(2),..., h(d))σh, is given by the formula: g h = (g h , g h , . . . , g h )σ σ . · (1) (σg(1)) (2) (σg(2)) (d) (σg(d)) g h g h = (g h , g h ,..., g h )σ σ . We define σ as (1, 2, . . .· , p) where(1) p(σisg(1 defined)) (2) by(σg( context.2)) (d) (σg(d)) g h The set X∗ is naturally a vertex set of a regular rooted tree, i.e. a connected graph without ( p) p cyclesWe and define a designatedσ as 1, 2, vertex . . . , vwhere0 called theis defined root, in by which context. two words are connected by an edge n if andThe only set ifX∗ theyis naturally are of form a vertexv and setvx of, a where regularv rootedX∗, x tree;X i.e.,. The a connected set X graphX∗ is without called cycles 0 ∈ ∈ j ⊂ andthe an-th designated level of the vertex treevX0 called∗ and theX root,= v0 in. which We denote two words by vji arethe connected vertex of X by, an which edge has if and the only if 2j { } n theynumber arei of, where form 1v andi vxX, whereand thev numerationX∗, x X. starts The set fromX 1. NoteX∗ is that called the the uniquen-th vertex level ofvk,i the tree ≤ ≤ ∈ ∈ ⊂ j Xcorrespondsand X0 = to thev . unique We denote word v byinv alphabetthe vertexX. For of X everyj, which automorphism has the numberg AutXi, where∗ and1 everyi X2 ∗ { 0} ji ∈ ≤ ≤ andword thev numerationX∗ determine starts the from section 1. Note (state) thatg(v the) uniqueAutX∗ vertexof g atvv bycorresponds the rule: g( tov)( thex) = uniquey for word ∈ ∈ k,i k i x, y X∗ if and only if g(vx) = g(v)y. The subtree of X∗ induced by the set of vertices i=0X v in alphabet∈ X.[ Fork] every automorphism g AutX∗ and every word v X∗ determine∪ the section is denoted by X . The restriction of the action∈ of an automorphism g AutX∈ ∗ to the subtree (state)[l] g(v) AutX∗ of g at v by the rule: g(v)(x) = y for x, y X∗ if and∈ only if g(vx) = g(v)y. The X is denoted∈ by g(v) [l] . A restriction g(v ) [1] is called the∈ vertex permutation (v.p.) of g at |X ij |Xk i [k] subtreea vertex ofvijXand∗ induced denoted by by thegij set. For of verticesexample, ifi=X0X =is 2 denotedthen we by justX have. The to distinguish restriction ofactive the action ∪ | [l]| ofvertices, an automorphism i.e., the verticesg forAutX whichtog theis subtree non-trivialX is [12]. denoted by g [l] . The restriction g [1] is ∈ ∗ ij (v)|X (vij)|X Let us label every vertex of Xl, 0 l < k by sign 0 or 1 in relation to state of v.p. in it. called the vertex permutation (v.p.) of g≤at a vertex vij and denoted by gij. For example, if X = 2 then Obtained by such way a vertex-labeled regular tree is an element of AutX[k]. All undeclared| | we just have to distinguish active vertices; i.e. the vertices for which gij is non-trivial [21]. terms are from [1,4]. We label every vertex of Xl, 0 l < k by 0 or 1 depending on the action of v.p. on it. The Let us fix some notations. For brevity,≤ in form of wreath[k] recursion we write a commutator resulting vertex-labeled1 1 regular tree is an element1 1 of AutX . All undeclared terms are from [22–24]. as [a, b] = aba− b− that is inverse to a− b− ab. That does not reduce the generality of our reasoning. Since for convenience the commutator of two group elements a and b is denoted by 1 1 [a, b] = aba− b− , conjugation by an element b as b 1 a = bab− , Mathematics 2020, 8, 472 3 of 19
Let us fix some notation. For convenience the commutator of two group elements a and b is 1 1 denoted by [a, b] = aba− b− , conjugation by an element b we denote by
b 1 a = bab− .
We define Gk and Bk recursively; i.e.,
B1 = C2, Bk = Bk 1 C2 for k > 1, − o G1 = e , Gk = (g1, g2)π Bk g1g2 Gk 1 for k > 1. h i { ∈ | ∈ − } k Note that Bk = C2. i=1o The commutator length of an element g of a derived subgroup of a group G, is the minimal n such that there exist elements x1, ... , xn, y1, ... , yn in G such that g = [x1, y1] ... [xn, yn]. 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(G) the minimal number of generators of the group G.
3. Commutator Width of Sylow 2-Subgroups of A2k and S2k 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 (r1, ... , rp 1, rp) W0 = (B Cp)0 iff product of all ri (in any order) belongs − ∈ o to B , where p N, p 2. 0 ∈ ≥ Proof. More details of our argument may be given as follows. If we multiply elements from a tuple 1 1 (r1,..., rp 1, rp) = w, where ri = hi ga(i)h− g− 1 , hi, gi B and a, b Cp, then we get a product − ab(i) aba− (i) ∈ ∈
p p 1 1 x = ri = hi ga(i)h− g− 1 B0, (1) ∏ ∏ ab(i) aba− (i) i=1 i=1 ∈
1 1 where x is a product of appropriate commutators. Therefore, we can write rp = r−p 1 ... r1− x. We can m − rewrite element x B0 as the product x = ∏ [hj, gj], m cw(B). ∈ j=1 ≤ Note that we impose a weaker condition on the product of all ri which belongs to B0 than in Definition 4.5 of form P(L) in [9], where the product of all ri belongs to a subgroup L of B such that L > B0. In more detail, deducing of our representation construct can be reported in the following way. 1 1 If we multiply elements having form of a tuple (r1, ... , rp 1, rp), where ri = hi ga(i)h− g− 1 , − ab(i) aba− (i) h , g B and a, b C , then we obtain a product i i ∈ ∈ p p p 1 1 ri = hi ga(i)h− g− 1 B0. (2) ∏ ∏ ab(i) aba− (i) i=1 i=1 ∈
1 1 1 1 1 1 Note that if we rearrange elements in (1) as h1h1− g1g2− h2h2− g1g2− ...hph−p gpg−p then by the reason of such permutations we obtain a product of appropriate commutators. Therefore, the following equality holds
p p p 1 1 1 1 1 1 hi ga(i)h− g− 1 = hi gih− g− x0 = hih− gi g− x B0, (3) ∏ ab(i) aba− (i) ∏ i i ∏ i i i=1 i=1 i=1 ∈ Mathematics 2020, 8, 472 4 of 19
where x0, x are the products of appropriate commutators. Therefore,
(r1,..., rp 1, rp) W0 iff rp 1 ... r1 rp = x B0. (4) − ∈ − · · · ∈
Thus, one element from states of wreath recursion (r1, ... , rp 1, rp) depends on rest of ri. This implies p − p that the product ∏ rj for an arbitrary sequence rj j=1 belongs to B0. Thus, rp can be expressed as: j=1 { }
1 1 rp = r1− ... r−p 1x. · · −
Denote a j-th tuple consisting of wreath recursion elements by (rj1 , rj2 , ..., rjp ). The fact that the set of forms (r1, ... , rp 1, rp) W = (B Cp)0 is closed under multiplication follows from the identity − ∈ o follows from
k k p (r r r ) = r = R R R B ∏ j1 ... jp 1 jp ∏ ∏ ji 1 2... k 0, (5) j=1 − j=1 i=1 ∈
p where rji is i-th element of the tuple number j, Rj = ∏ rji, 1 j k. As it was shown above i=1 ≤ ≤ p 1 − Rj = ∏ rji B0. Therefore, the product (5) of Rj, j 1, ..., k which is similar to the product i=1 ∈ ∈ { } mentioned in [9], has the property R R ...R B too, because of B is subgroup. Thus, we get a 1 2 k ∈ 0 0 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 ri, where every i occurs in this form once, belong to B0. Then using the elements of the form 1 1 1 (r1, e, ..., e, r1− ), ... , (e, e, ..., e, ri, e, ri− ), ... ,(e, e, ..., e, rp 1, r−p 1), (e, e, ..., e, r1r2 ... rp 1) − − · · − we can express any elements of the form (r1, ... , rp 1, rp) W = (B Cp)0. We need to prove that − ∈ o 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 ri, i < p are arbitrary and the fact that equality (1) holds, so rp is well determined.
Lemma 2. Assume a group B and an integer p 2. If w (B C ) then w can be represented as the following ≥ ∈ o p 0 wreath recursion
k 1 1 w = (r1, r2,..., rp 1, r1− ... r−p 1 ∏[ fj, gj]), − − j=1 where r1,..., rp 1, fj, gj B and k cw(B). − ∈ ≤ Proof. According to Lemma1 we have the following wreath recursion
w = (r1, r2,..., rp 1, rp), − 1 1 where ri B and rp 1rp 2 ... r2r1rp = x B0. Therefore, we can write rp = r1− ... r−p 1x. We can also ∈ − − ∈ − k rewrite an element x B0 as a product of commutators x = ∏ [ fj, gj] where k cw(B). ∈ j=1 ≤
Lemma 3. For any group B and integer p 2, suppose w (B C ) is defined by the following wreath ≥ ∈ o p 0 recursion:
1 1 w = (r1, r2,..., rp 1, r1− ... r−p 1[ f , g]), − − Mathematics 2020, 8, 472 5 of 19
where r1,..., rp 1, f , g B. Then we can represent w as the following commutator − ∈
w = [(a1,1,..., a1,p)σ, (a2,1,..., a2,p)], where
a = e, for 1 i p 1 , 1,i ≤ ≤ − 1 1 1 r1− ...r−p 1 a2,1 = ( f − ) − ,
a2,i = ri 1a2,i 1, for 2 i p, − − ≤ ≤ 1 a2,−p a1,p = g .
Proof. Consider the following commutator
1 1 1 1 1 1 κ = (a1,1,..., a1,p)σ (a2,1,..., a2,p) (a1,−p, a1,1− ,..., a1,−p 1)σ− (a2,1− ,..., a2,−p) · · − · = (a3,1,..., a3,p), where
1 1 a3,i = a1,ia2,1+(i mod p)a1,−i a2,−i .
At first we compute the following
1 1 1 1 a = a a a− a− = a a− = r a a− = r , for 1 i p 1. 3,i 1,i 2,i+1 1,i 2,i 2,i+1 2,i i 2,i 2,i i ≤ ≤ −
Then we make some transformation of a3,p:
1 1 a3,p = a1,pa2,1a1,−pa2,−p 1 1 1 = (a2,1a2,1− )a1,pa2,1a1,−pa2,−p 1 1 = a2,1[a2,1− , a1,p]a2,−p 1 1 1 = a2,1a2,−pa2,p[a2,1− , a1,p]a2,−p a 1 1 1 a2,p 2,p = (a2,pa2,1− )− [(a2,1− ) , a1,p ]
1 a2,p 1 1 1 a2,p a2,1− = (a2,pa2,1− )− [(a2,1− ) , a1,p ].
Now we can see that the form of the commutator κ is similar to the form of w. Introduce the following notation
r0 = rp 1 ... r1. − We note that from the definition of a for 2 i p it follows that 2,i ≤ ≤ 1 r = a a− , for 1 i p 1. i 2,i+1 2,i ≤ ≤ − Therefore
1 1 1 1 r0 = (a2,pa2,−p 1)(a2,p 1a2,−p 2) ... (a2,3a2,2− )(a2,2a2,1− ) − − − 1 = a2,pa2,1− .
Then 1 1 1 1 1 (a2,pa2,1− )− = (r0)− = r1− ... r−p 1. − Mathematics 2020, 8, 472 6 of 19
Now we compute the following
1 r 1...r 1 1 1 a2,p a2,1− 1 1− −p 1 1 r0 (r0)− r0 (a2,1− ) = ((( f − ) − )− ) = ( f ) = f , a 1 2,p a2,−p a2,p a1,p = (g ) = g.
Finally, we conclude that
1 1 a3,p = r1− ... r−p 1[ f , g]. − Thus, the commutator κ 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 (B C ) is defined by the following wreath recursion ∈ o 2 0 1 w = (r1, r1− [ f , g]), where r , f , g B. Then we can represent w as commutator 1 ∈
w = [(e, a1,2)σ, (a2,1, a2,2)], where
1 1 r− a2,1 = ( f − ) 1 ,
a2,2 = r1a2,1, 1 a− a1,2 = g 2,2 .
Lemma 4. For any group B and integer p 2 the inequality ≥ cw(B C ) max(1, cw(B)) o p ≤ holds.
Proof. By Lemma1, we can represent any w (B C ) as the following wreath recursion ∈ o p 0 k 1 1 w = (r1, r2,..., rp 1, r1− ..., r−p 1 ∏[ fj, gj]) − − j=1 k 1 1 = (r1, r2,..., rp 1, r1− ..., r−p 1[ f1, g1]) ∏[(e,..., e, fj), (e,..., e, gj)], − − · j=2 where r1, ... , rp 1, fj, gj B and k cw(B). Now by the Lemma3 we can see that w can be represented − ∈ ≤ as a product of max(1, cw(B)) commutators.
Corollary 2. If W = C ... C then cw(W) = 1 for k 2. pk o o p1 ≥
Proof. If B = Cpk Cpk 1 , then take into consideration that cw(B) > 0 (because Cpk Cpk 1 is not o − o − commutative group). Lemma4 implies that cw(Cpk Cpk 1 ) = 1, and using the inequality cw(Cpk o − o Cpk 1 Cpk 2 ) max(1, cw(B)) from Lemma4 we obtain cw(Cpk Cpk 1 Cpk 2 ) = 1. Similarly, if − o − ≤ o − o − W = C ... C we use inductive assumption for C ... C the associativity of a permutational pk o o p1 pk o o p2 wreath product, the inequality of Lemma4 and the equality cw(C ... C ) = 1 to conclude that pk o o p2 cw(W) = 1. Mathematics 2020, 8, 472 7 of 19
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 N.
k Hk = pi (6) i=o1 C
k Moreover, it has already been proved in Corollary3 that each group of the form pi has a i=o1 C k commutator width equal to 1; i.e., cw( pi ) = 1. A partially ordered set of a subgroups is ordered by i=o1 C k relation of inclusion group as a subgroup. Define the injective homomorphism fk,k+1 from the pi i=o1 C k+1 into pi by mapping a generator of active group pi of Hk in a generator of active group pi of Hk+1. i=o1 C C C In more detail, the injective homomorphism fk,k+1 is defined as g g(e, ..., e), where a generator k k+1 7→ g pi , g(e, ..., e) pi . ∈ i=o1 C ∈ i=o1 C k We therefore obtain an injective homomorphism from Hk onto the subgroup pi of Hk+1. i=o1 C
k k Corollary 3. The direct limit lim of the direct system f , has commutator width 1. o Cpi k,j o Cpi −→i=1 i=1
k Proof. We make the transition to the direct limit in the direct system f , of injective k,j o Cpi i=1 k k+1 k+2 mappings from chain e ... pi pi pi .... → → i=o1 C → i=o1 C → i=o1 C → 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(H) = 1, because each group Hk from the chain is endowed by cw(Hk) = 1. k The direct limit of the direct system is denoted by lim pi and is defined as disjoint union of −→i=o1 C the Hk’s modulo a certain equivalence relation:
k k ä pi k i=o 1 C lim pi = / . −→i=o1 C ∼ k Since every element g of lim pi coincides with a correspondent element from some Hk of direct −→i=o1 C k system, then by the injectivity of the mappings for g the property cw( pi ) = 1 also holds. Thus, it i=o1 C k holds for the whole lim pi . −→i=o1 C
Corollary 4. For prime p and k 2 we have cw(Syl (S k )) = 1. For prime p > 2 and k 2 we have ≥ p p ≥ cw(Sylp(Apk )) = 1.
k Proof. Since Sylp(Spk ) Cp (see [25,26]), we have cw(Sylp(Spk )) = 1. It is well known that in a 'i=1o case p > 2 where we have Syl S k Syl A k (see [15,23]), so we obtain cw(Syl (A k )) = 1. p p ' p p p p
Proposition 1. There is an inclusion Bk0 < Gk holds. Mathematics 2020, 8, 472 8 of 19
Proof. We use induction on k. For k = 1 we have B = G = e . Fix some g = (g , g ) B . Then k0 k { } 1 2 ∈ k0 g1g2 Bk0 by Lemma1. As Bk0 < Gk 1 by the induction hypothesis therefore g1g2 Gk 1 and by ∈ 1 1 − ∈ − definition− of G it follows that g− G . k ∈ k
Corollary 5. The set Gk is a subgroup in the group Bk.
Proof. According to the recursively definition of G and B , where G = (g , g )π B g g k k k { 1 2 ∈ k | 1 2 ∈ Gk 1 k > 1, i.e. Gk is subset of Bk with condition g1g2 Gk 1. The result follows from the fact that − } ∈ − Gk 1 is a subgroup of Gk. It is easy to check the closedness by multiplication elements of Gk with − condition g1g2, h1h2 Gk 1 because Gk 1 is subgroup so g1g2h1h2 Gk 1 too. The inverses can be ∈ − − ∈ − verified easily.
Lemma 5. For any k 1 we have G = B /2. ≥ | k| | k| Proof. Induction on k. For k = 1 we have G = 1 = B /2 . Every element g G can be uniquely | 1| | 1 | ∈ k written as the following wreath recursion
1 g = (g1, g2)π = (g1, g1− x)π where g1 Bk 1, x Gk 1 and π C2. Elements g1, x and π are independent; therefore, Gk = ∈ − ∈ − ∈ | | 2 Bk 1 Gk 1 = 2 Bk 1 Bk 1 /2 = Bk /2. | − | · | − | | − | · | − | | |
Corollary 6. The group Gk is a normal subgroup in the group Bk; i.e., Gk / Bk.
Proof. There exists normal embedding (normal injective monomorphism) ϕ : G B [27] such k → k that G / B . Indeed, according to Lemma index B : G = 2, so it is a normal subgroup; that is, a k k | k k| quotient subgroup Bk / G . C2 ' k
Theorem 1. For any k 1 we have G Syl A k . ≥ k ' 2 2 k Proof. Group C2 acts on the set X = 1, 2 . Therefore, we can recursively define sets X on which 1 k k 1 { } k k group B acts X = X, X = X X for k>1. At first we define S k = Sym(X ) and A k = Alt(X ) k − × 2 2 for all integers k 1. Then G < B < S k and A k < S k . ≥ k k 2 2 2 We already know [15] that Bk Syl2(S2k ). Since A2k = S2k /2, Syl2 A2k = Syl2S2k /2 = ' | | | | | | | | k Bk /2. By Lemma3 it follows that Syl2 A2k = Gk . Therefore, it remains to show that Gk < Alt(X ). | | i| | | | Let us fix some g = (g1, g2)σ where g1, g2 Bk 1, i 0, 1 and g1g2 Gk 1. Then we can ∈ − ∈ { } ∈ − represent g as follows 1 i g = (g g , e) (g− , g ) (e, e, )σ . 1 2 · 2 2 · 1 k In order to prove this theorem it is enough to show that (g1g2, e), (g2− , g2), (e, e, )σ Alt(X ). k ∈ Elements (e, e, )σ just switch letters x1 and x2 for all x X . Therefore, (e, e, )σ is product of k 1 k 1 k∈ X − = 2 − transpositions, and therefore, (e, e, )σ Alt(X ). | | 1 ∈ 1 Elements g2− and g2 have the same cycle type. Therefore, elements (g2− , e) and (e, g2) also have the same cycle type. Let us fix the following cycle decompositions
1 (g− , e) = σ ... σ , 2 1 · · n (e, g ) = π ... π . 2 1 · · n 1 Note that element (g2− , e) acts only on letters like x1, and element (e, g2) acts only on letters like x2. Therefore, we have the following cycle decomposition
1 (g− , g ) = σ ... σ π ... π . 2 2 1 · · n · 1 · · n Mathematics 2020, 8, 472 9 of 19
1 1 k So, element (g2− , g2) has even number of odd permutations and then (g2− , g2) Alt(X ). k 1 ∈ Note that g1g2 Gk 1 and Gk 1 = Alt(X − ) by induction hypothesis. Therefore, g1g2 ∈ − − ∈ Alt(Xk 1). As elements g g and (g g , e) have the same cycle type, (g g , e) Alt(Xk). − 1 2 1 2 1 2 ∈
AsAs it itwas was proven proven by by the the author author in [ [19]15], the Sylow Sylow 2-subgroup 2-subgroup has has structure structureBBkk 11 n Wkk 11,, where where −− −− definitionthe definition of B ofk B1kis1 theis the same same that that was which given was in given [19]. in [15]. − − [k] RecallRecall that that it it was was denoted denoted by WWk 1thethe subgroup subgroup of ofAutXAutX[k] suchsuch that that it had has active active states states only only on k −1 kX1k 1 − W St (k 1) onX − −andand a number number of of such such states states that is was even, even; i.e. i.e.,k W1k▹1 / StGkG (k 1)[12].[21]. It It was was proven proven that that the the k 1 − k −k 1 k 21 − 1 − k 1 2−− 1 size of WWk 1 is equal to 22− 1−k,> k > 1 and its structure(C ) is2 −(C21) − . The following structural size of k−1 is equal to 2 − , 1 and its structure is 2 − . The following structural theorem theorem characterizing− the group G was proved by us [19]. characterizing the group Gk was provenk by us [15]. Theorem 14. A maximal 2-subgroup of AutX[k] that acts by even permutations on Xk has the [k] k structureTheorem of 2. theA maximal semidirect 2-subgroup product of GAutXk Bkthat1 n actsW byk even1 and permutations isomorphic on toX Sylhas2 theA2k structure. of the ≃ − − semidirect product Gk Bk 1 Wk 1 and isomorphic tok Syl1 2 A2k . [k] Note that Wk 1 is' subgroup− of− stabilizer of X − i.e. Wk 1 < StAutX[k] (k 1) ▹ AutX and − [k] − − is normal too Wk 1 ▹ AutX , because conjugationk 1 keeps a cyclic structure of permutation[k] so Note that Wk− 1 is subgroup of stabilizer of X − , i.e., Wk 1 < StAutX[k] (k 1) / AutX and even permutation− maps in[k] even. Therefore such conjugation− induce an automorphism− of Wk 1 is normal to Wk 1 / AutX , because conjugation keeps a cyclic structure of permutation, so even− and Gk Bk 1 n− Wk 1. permutation≃ − maps are− even. Therefore, such conjugation induce an automorphism of Wk 1 and Remark 15. As a consequence, the structure founded by us in [19] fully consistent with− the Gk Bk 1 Wk 1. recursive' − group representation− (which used in this paper) based on the concept of wreath recursion [8].Remark 1. As a consequence, the structure founded by us in [15] is fully consistent with the recursive group Theoremrepresentation 16. (usedElements in this paper) of B′ basedhave on the the following concept of wreath form B recursion′ = [f, [10 l] ]. f Bk, l Gk = [l, f] k k { | ∈ ∈ } { | f B , l G . ∈ k ∈ k} Theorem 3. Elements of Bk0 have the following form Bk0 = [ f , l] f Bk, l Gk = [l, f ] f Bk, l Proof. It is enough to show either Bk′ = [f, l] f B{k, l |Gk∈ or B∈k′ = }[l, f{] f | Bk∈, l G∈k Gk . 1 { | ∈ ∈ } { | ∈ ∈ } because} if f = [g, h] then f − = [h, g]. We prove the proposition by induction on k. For the case k = 1 we have B1′ = e . Proof. It is enough to show either Bk0 = [ f , l] f Bk, l Gk or Bk0 = [l, f ] f ⟨ B⟩k, l Gk , Consider case k > 1. According to Lemma{ | 2 and∈ Corollary∈ } 4 every{ element| w∈ B′ ∈can} be because if f = [g, h], then f 1 = [h, g]. ∈ k represented as − We prove the proposition by induction on k. For the case k = 1 we have B0 = e . 1 1 h i Consider case k > 1. According tow Lemma= (r1,2 r1 −and[f, Corollary g]) 1 every element w B can be ∈ k0 represented as for some r1, f Bk 1 and g Gk 1 (by induction hypothesis). By the Corollary 4 we can ∈ − ∈ − represent w as commutator of 1 w = (r1, r1− [ f , g]) (e, a )σ B and (a , a ) B , 1,2 ∈ k 2,1 2,2 ∈ k for some r1, f Bk 1 and g Gk 1 (by induction hypothesis). By the Corollary1 we can represent w where ∈ − ∈ − as commutator of 1 1 r− a2,1 = (f − ) 1 , (e, a1,2)σ Bk and (a2,1, a2,2) Bk, a∈2,2 = r1a2,1, ∈ 1 a− where a1,2 = g 2,2 .
1 If g Gk 1 then by the definition of Gk and Corollary1 r− 12 we obtain (e, a1,2)σ Gk. ∈ − a2,1 = ( f − ) 1 , ∈ Remark 17. Let us to note that Theorema2,2 16= improver1a2,1, Corollary 8 for the case Syl2S2k . 1 2 Proposition 18. If g is an element of the groupa2,2− Bk then g B′ . a1,2 = g . ∈ k Proof. Induction on k. We note that Bk = Bk 1 C2. Therefore we fix some element If g Gk 1, then by the definition of Gk and Corollary− ≀ 6 we obtain (e, a1,2)σ Gk. ∈ − i ∈ g = (g1, g2)σ Bk 1 C2, ∈ − ≀ Remark 2. Let us to note that Theorem3 improve Corollary4 for2 the case Syl 2S2k . where g1, g2 Bk 1 and i 0, 1 . Let us to consider g then two cases are possible: ∈ − ∈ { } Proposition 2. If g is an element ofg2 the= group (g2, g B2)thenor g g22= (Bg .g , g g ). 1 2k ∈ k01 2 2 1 In second case we consider a product of coordinates g g g g = g2g2x. Since according to the 1 2 · 2 1 1 2 induction hypothesis g2 B , i 2 then g g g g B also according to Lemma 1 x B . i ∈ k′ ≤ 1 2 · 2 1 ∈ k′ ∈ k′ Therefore a following inclusion holds (g g , g g ) = g2 B . In first case the proof is even 1 2 2 1 ∈ k′ simpler because g2, g2 B by the induction hypothesis. 1 2 ∈ ′ Mathematics 2020, 8, 472 10 of 19
Proof. Induction on k. We note that Bk = Bk 1 C2. Therefore, we fix some element − o i g = (g1, g2)σ Bk 1 C2, ∈ − o 2 where g1, g2 Bk 1 and i 0, 1 . Let us to consider g . Then, two cases are possible: ∈ − ∈ { } 2 2 2 2 g = (g1, g2) or g = (g1g2, g2g1).
In the second case we consider a product of coordinates g g g g = g2g2x. Since according to the 1 2 · 2 1 1 2 induction hypothesis g2 B , i 2 then g g g g B also according to Lemma1 x B . Therefore, i ∈ k0 ≤ 1 2 · 2 1 ∈ k0 ∈ k0 a following inclusion holds (g g , g g ) = g2 B . In first case the proof is even simpler because 1 2 2 1 ∈ k0 g2, g2 B by the induction hypothesis. 1 2 ∈ 0
Lemma 6. If an element g = (g1, g2) Gk0 then g1, g2 Gk 1 and g1g2 Bk0 1. ∈ ∈ − ∈ −
Proof. As Bk0 < Gk, it is therefore enough to show that g1 Gk 1 and g1g2 Bk0 1. Let us fix some ∈ − ∈ − g = (g1, g2) Gk0 < Bk0 . Then, Lemma1 implies that g1g2 Bk0 1. ∈ ∈ − In order to show that g1 Gk 1, we firstly consider just one commutator of arbitrary elements ∈ − from Gk
f = ( f , f )σ, h = (h , h )π G , 1 2 1 2 ∈ k where f1, f2, h1, h2 Bk 1, σ, π C2. The definition of Gk implies that f1 f2, h1h2 Gk 1. ∈ − ∈ ∈ − If g = (g1, g2) = [ f , h], then 1 1 g1 = f1hi fj− hk− for some i, j, k 1, 2 . Then ∈ { } 1 2 1 2 1 1 2 g1 = f1hi fj( fj− ) hk(hk− ) = ( f1 fj)(hihk)x( fj− hk− ) , where x is product of commutators of fi, hj and fi, hk; hence, x Bk0 1. ∈ 2− It is enough to consider the first product f1 fj. If j = 1, then f1 Bk0 1 by Proposition2 if j = 2 ∈ − then f1 f2 Gk 1 according to definition of Gk; the same is true for hihk. Thus, for any i, j, k it holds ∈ − 1 1 2 that f1 fj, hihk Gk 1. Besides that, a square ( fj− h− ) Bk0 according to Proposition2. Therefore, ∈ − k ∈ g1 Gk 1 because of Propositions1 and2, the same is true for g2. ∈ − Now it remains to consider the product of some f = ( f1, f2), h = (h1, h2), where f1, h1 Gk 1, ∈ − f1h1 Gk 1 and f1 f2, h1h2 Bk0 1 ∈ − ∈ −
f h = ( f1h1, f2h2).
Since f1 f2, h1h2 Bk0 1 by imposed condition in this item and taking into account that f1h1 f2h2 = ∈ − f1 f2h1h2x for some x Bk0 1, then f1h1 f2h2 Bk0 1 by Lemma1. In other words, closedness by ∈ − ∈ − multiplication holds, and so according to Lemma1, we have element of commutator Gk0 . In the following theorem we prove two facts at once.
Theorem 4. The following statements are true.
1. An element g = (g1, g2) Gk0 iff g1, g2 Gk 1 and g1g2 Bk0 1. ∈ ∈ − ∈ − 2. Commutator subgroup G coincides with set of all commutators for k 3 k0 ≥
G0 = [ f , f ] f G , f G . k { 1 2 | 1 ∈ k 2 ∈ k} Mathematics 2020, 8, 472 11 of 19
Proof. For the case k = 1 we have G = e . So, further we consider the case k 2. If k = 2 then we 10 h i ≥ have G V , where V is the Klein four group. But cw(V ) = 0. 2 ' 4 4 4 Sufficiency of the first statement of this theorem follows from the Lemma6. So, in order to prove the necessity of the both statements it is enough to show that element
1 w = (r1, r1− x), where r1 Gk 1 and x Bk0 1, can be represented as a commutator of elements from Gk. By ∈ − ∈ − Proposition3 we have x = [ f , g] for some f Bk 1 and g Gk 1. Therefore, ∈ − ∈ − 1 w = (r1, r1− [ f , g]).
By the Corollary1 we can represent w as a commutator of
(e, a )σ B and (a , a ) B , 1,2 ∈ k 2,1 2,2 ∈ k
1 1 1 r− a− where a2,1 = ( f − ) 1 , a2,2 = r1a2,1, a1,2 = g 2,2 . It only remains to show that (e, a1,2)σ, (a , a ) G . Note the following 2,1 2,2 ∈ k 1 a− a1,2 = g 2,2 Gk 1 by Corollary6. ∈ − 2 a2,1a2,2 = a2,1r1a2,1 = r1[r1, a2,1]a2,1 Gk 1 by Propositions1 and2. ∈ − So we have (e, a )σ G and (a , a ) G by the definition of G . 1,2 ∈ k 2,1 2,2 ∈ k k Proposition 3. For arbitrary g G the inclusion g2 G holds. ∈ k ∈ k0 2 2 Proof. Induction on k: elements of G1 have form (σ) = e, where σ = (1, 2), so the statement holds. In i a general case, when k > 1, the elements of Gk have the form g = (g1, g2)σ , g1, g2 Bk 1, i 0, 1 . 2 2 2 2 ∈ − ∈ { } Then we have two possibilities: g = (g1, g2) or g = (g1g2, g2g1). 2 2 2 2 Firstly we show that g1 Gk 1, g2 Gk 1. According to Proposition2, we have g1, g2 Bk0 1 ∈ − ∈ − 2 2 2 ∈ − and according to Proposition1, we have Bk0 < Gk 1. Then, using Theorem4 g = (g1, g2) Gk. 1 − ∈ Consider the second case g2 = (g g , g−g ). Since g G , then, according to the definition of G , 1 2 2 1 ∈ k k we have that g1g2 Gk 1. By Proposition1, and the definition of Gk, we obtain ∈ − 1 1 1 1 g2g1 = g1g2g2− g1− g2g1 = g1g2[g2− , g1− ] Gk 1, ∈ − 2 2 2 2 1 g1g2 g2g1 = g1g2g1 = g1g2[g2− , g1− ] Bk0 1. · ∈ − 2 2 2 2 2 1 Note that g1, g2 Bk0 1 according to Proposition2, g1g2[g2− , g1− ] Bk0 1. Since g1g2 g2g1 Bk0 1 ∈ − 2 ∈ − · ∈ − and g1g2, g2g1 Gk 1, then, according to Lemma6, we obtain g = (g1g2, g2g1) Gk0 . ∈ − ∈ 2 2 Statement 1. The commutator subgroup is a subgroup of Gk ; i.e., G0k < Gk .
Proof. Indeed, an arbitrary commutator presented as the product of squares. Let a, b G and set 1 1 1 2 2 2 2 1 2 1 1 2 1 1 ∈ that x = a, y = a− ba, z = a− b− . Then x y z = a (a− ba) (a− b− ) = aba− b− . In more detail: 2 1 2 1 1 2 2 1 1 1 1 1 1 1 1 1 a (a− ba) (a− b− ) = a a− ba a− ba a− b− a− b− = abbb− a− b− = [a, b]. In such way we obtain all commutators and their products. Thus, we generate by squares the whole G0k.
2 Corollary 7. For the Syllow subgroup (Syl2 A2k ) the following equalities Syl20 (A2k ) = (Syl2(A2k )) , Φ(Syl2 A2k ) = Syl20 (A2k ), which are characteristic properties of special p-groups [28], are true. Mathematics 2020, 8, 472 12 of 19
2 Proof. As is well known, for an arbitrary group (also by Statement1) the following embedding G0 / G holds. In view of the above Proposition3, a reverse embedding for Gk is true. Thus, the group 2 Syl2 A2k has some properties of special p-groups; that is, P0 = Φ(P) [28] because Gk = Gk0 and so Φ(Syl2 A2k ) = Syl20 (A2k ).
Corollary 8. Commutator width of the group Syl A k is equal to 1 for k 3, also cw(Syl A ) = 0. 2 2 ≥ 2 4 It immediately follows from item 2 of Theorem4 and the fact that Syl A V . 2 4 ' 4 4. Minimal Generating Set For the construction of minimal generating set, we used the representation of elements of group k Gk by portraits of automorphisms at restricted binary tree AutX . For convenience we will identify [k] elements of Gk with their faithful representations by portraits of automorphisms from AutX . We denote by A , a set of all functions a , such that [ε, ... , ε, a , ε, ...] [A] . Recall that according |l l l ∈ l to [29], l-coordinate subgroup U < G is the following subgroup.
Definition 1. For an arbitrarry k N we call a k coordinate subgroup U < G a subgroup, which is ∈ − determined by k-coordinate sets [U]l, l N, if this subgroup consists of all Kaloujnine’s tableaux a I for ∈ ∈ which [a] [U] . l ∈ l We denote by G (l) a level subgroup of G , which consists of the tuples of v.p. from Xl, l < k 1 of k k − any α Gk. We denote as Gk(k 1) such subgroups of Gk that are generated by v.p., which are located k ∈1 − on X − and isomorphic to Wk 1. Note that Gk(l) is in bijective correspondence (and isomorphism) − with l-coordinate subgroup [U]l [29]. For any v.p. g in v of Xl we set in correspondence with g the permutation ϕ (g ) S by the li li li li ∈ 2 following rule:
(1, 2), if gli = e, ϕ(gli) = 6 (7) ( e, if gli = e.
Define a homomorphic map from Gk(l) onto S2 with the kernel consisting of all products of even number of transpositions that belong to Gk(l). For instance, the element (12)(34) of Gk(2) belongs to kerϕ. Hence, ϕ (g ) S . li ∈ 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 Gk0, the following k equalities are true:
2l ∏ ϕ(glj) = e, 0 l < k 1. (8) l=1 ≤ −
For the case i = k 1, the following condition holds: −
k 2 k 1 2 − 2 − ∏ ϕ(gk 1j) = ∏ ϕ(gk 1j) = e. (9) − k 2 − j=1 j=2 − +1
Thus, G0k has k new conditions on a combination of level subgroup elements, except for the condition of last level parity from the original group. 4. Minimal generating set For the construction of minimal generating set we used the representation of elements of group Gk by portraits of automorphisms at restricted binary tree AutXk. For convenience we will identify [k] elements of Gk with its faithful representation by portraits of automorphisms from AutX . We denote by A a set of all functions a , such, that [ε, . . . , ε, a , ε, . . .] [A] . Recall that, |l l l ∈ l according to [21], l-coordinate subgroup U < G is the following subgroup. Definition 1. For an arbitrarry k N we call a k coordinate subgroup U < G a subgroup, ∈ − which is determined by k-coordinate sets [U]l, l N, if this subgroup consists of all Kaloujnine’s ∈ tableaux a I for which [a] [U] . ∈ l ∈ l l We denote by Gk(l) a level subgroup of Gk, which consists of the tuples of v.p. from X , l < k 1 of any α Gk. We denote as Gk(k 1) such subgroup of Gk that is generated − ∈ k 1 − by v.p., which are located on X − and isomorphic to Wk 1. Note that Gk(l) is in bijective − correspondence (and isomorphism) with l-coordinate subgroup [U]l [21]. For any v.p. g in v of Xl we set in correspondence with g the permutation φ (g ) S by li li li li ∈ 2 the following rule:
(1, 2), if gli = e, φ(gli) = ̸ (7) e, if g = e. { li Define a homomorphic map from Gk(l) onto S2 with the kernel consisting of all products of even number of transpositions that belongs to Gk(l). For instance, the element (12)(34) of Gk(2) belongs to kerφ. Hence, φ (g ) S . li ∈ 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 Gk′, the following k equalities are true:
2l φ(g ) = e, 0 l < k 1. (8) lj ≤ − ∏l=1 For the case i = k 1, the following condition holds: −
k 2 k 1 2 − 2 − φ(gk 1j) = φ(gk 1j) = e. (9) − − j=1 k 2 ∏ j=2∏− +1
Thus,MathematicsG′k has2020,k8,new 472 conditions on a combination of level subgroup elements, except for13 the of 19 condition of last level parity from the original group.
Proof.Proof.NoteNote that that the the condition condition (8) (8 is) is compatible compatible with with thosethat were which founded were founded by R. byGuralnik R. Guralnik in [27], in [8], k 2k 2 − − (i) (i) becausebecause as it as was it was proved proven by byauthor author [19] [15G] Gk k1 1 BBkk 22 o k 11,, where whereBBk k2 2 C2 C. 2 . −− ≃' −− W −− − −' ≃i=o1i=1≀ 2 2 AccordingAccording to Property to Property 1, G1′k, G0Gk k,G sok , it so is it enough is enough to to prove prove the the statement statement for for the the elements elements of 2 ≤ ≤ 2 l of GkG.k Such. Such elements, elements, as as it it was was described described above, can can be be presented presented in in the the form forms =s = (sl1 (,s ...,l1,s ...,l2l ) sσl2, where)σ, σ G s s G lv i 2l whereσ Gl 1land1 andsli areli statesare states of s ofGk in vli, ki in2 .li For, convenience. For convenience we will make we the will transition make the from ∈ ∈ − − ∈ ∈ ≤ ≤ transition from the tuple (sl1, ..., s l ) to the tuple (gl1, ..., g l ). Note that there is the trivial2 the tuple (sl1, ..., sl2l ) to the tuplel2(gl1, ..., gl2l ). Note that therel2 is the trivial vertex permutation glj = e vertex permutation g2 = e in the product of the states s s . in the product of thelj states slj slj. lj · lj 0 0 · Since inSinceG′k inv.p.G0k onv.p.X onareX trivial,are trivial, so σ socanσ can be be decomposed decomposed as asσ σ== (σ(11σ11,, σσ2121)),, where where σ21,,σ σ2222are are rootroot permutations permutations in invv1111andandv12v12. . 2 Consider the square of s. We calculate squares ((s , s , ..., s l 1 ) σ) . The condition (8) is l1 l2 l2 − equivalent to the condition that s2 has even index on each level. Two cases are feasible: if permutation 2 2 2 2 2 2 2 σ = e, then ((sl1, sl2, ..., sl l 1 ) σ) = s , s , ..., s l 1 e, so after the transition from s , s , ..., s l 1 2 − l1 l2 l2 − l1 l2 l2 − 2 2 2 l 2 to g , g , ..., g l 1 , we get a tuple of trivial permutations (e, ... , e) on X , because g = e. In the l1 l2 l2 − lj 2 general case, if σ = e, after such transition we obtain gl1g ( ), ... , g l 1 g l 1 σ . Consider the 6 lσ 2 l2 − lσ(2 − ) product of form
2l ∏ ϕ(gljglσ(j)), (10) j=1
2 where σ and g g are from g g , ... , g l 1 g l 1 σ . li lσ(i) l1 lσ(2) l2 − lσ(2 − ) Note that each element glj occurs twice in (10) regardless of the permutation σ; therefore, considering the commutativity of homomorphic images ϕ(g ), 1 j 2l we conclude that lj ≤ ≤ 2l 2l 2l 2 2 2 ∏ ϕ(gljglσ(j)) = ∏ ϕ(glj) = e, because of glj = e. We rewrite ∏ ϕ(glj) = e as characteristic condition: j=1 j=1 j=1 l 1 l 2 − 2 ∏ ϕ(glj) = ∏ ϕ(glj) = e. l 1 j=1 j=2 − +1 According to Property1, any commutator from G0k can be presented as a product of some squares 2 s , s G , s = ((s , ..., s l )σ ). ∈ k l1 l2 2l A product of elements of Gk(k 1) satisfies the equation ∏ ϕ(glj) = e, because any permutation − j=1 k of elements from X , which belongs to Gk is even. Consider the element s = (sk 1,1, ..., sk 1,2k 1 )σ, − − − where (s , ..., s k 1 ) G (k 1), σ G . If g = (1, 2), where g is root permutation of σ, k 1,1 k 1,2 − k k 1 01 01 2 − − ∈ − ∈ − k 1 k 1 k 1 then s = (sk 1,1sk 1σ(1), ..., sk 1,(2k 1)sk 1,σ(2k 1)), where σ(j) > 2 − for j 2 − . And if j < 2 − then − − − − − − ≤ 2k 1 k 1 − k 1 k 1 σ(j) 2 − . Because of ∏ ϕ(gk 1,j) = e holds in Gk and the property σ(j) 2 − hold for j > 2 − , ≥ j=1 − ≤ k 2 2 − then the product ∏ ϕ(gk 1,jgk 1,σ(j)) of images of v.p. from (gk 1,1gk 1,σ(1), ..., gk 1,(2k 1)gk 1,σ(2k 1)) j=1 − − − − − − − − k 1 k 1 k 1 2 − 2 − 2 − is equal to ∏ ϕ(gk 1,j) = e. Indeed, the products ∏ ϕ(gk 1,j) and ∏ ϕ(gk 1,jgk 1,σ(j)) have the j=1 − j=1 − j=1 − − k 1 same v.p. from X − which do not depend on such σ as described above. k 1 The same is true for right half of X − . Therefore, the equality (9) holds. k 1 2 − Note that such product ∏ ϕ(gk 1,j) is homomorphic image of (gl,1gl,σ(1), ..., gl,(2l )glσ(2l )), where j=1 − l = k 1, as an element of G (l) after mapping (7). − k0 If g01 = e, where g01 is root permutation of σ, then σ can be decomposed as σ = 2 (σ11, σ12), where σ11, σ12 are root permutations in v11 and v12. As a result s has a form Mathematics 2020, 8, 472 14 of 19
2 2 (sl1s ( ), ..., s l 1 )σ , (s l 1 s l 1 , ..., s l s l )σ , where l = k 1. As a result of action lσ 1 lσ(2 − ) 1 l2 − +1 lσ(2 − +1) l(2 ) lσ(2 ) 2 − of σ all states of l-th level with number 1 j 2k 2 permutes in the set of coordinate from 1 to 2k 2. 11 ≤ ≤ − − The others are fixed. The action of σ11 is analogous. It corresponds to the next form of element from G (l): (g g , ..., g l 1 ), k0 l1 lσ1(1) lσ1(2 − ) k 2 2 − k 2 (g l 1 g l 1 , ..., g l g l ). Therefore, the equality of2 form− ∏ ϕ(g g ) = l2 − +1 lσ2(2 − +1) l(2 ) lσ2(2 ) k 1,j lσ(j) (g l 1 g l 1 , ..., g l g l ) φ(gj=1 g − ) = l2 − +1 lσ2(2 − +1) l(2 ) lσ2(2 ) . Therefore the product of form k 1,j lσ(j) k 1 j=1 − 2 − k 1 2 2 2 − ∏ ϕ(gk 1,j) = e, because of gk 1,j = e holds. Thus, characteristic∏ equation (9) of k 1 level k 2 − 2 − − j=φ2(g−k+11,j) = e, because of gk 1,j = e. Thus, characteristic equation (9) of k 1 level k 2 − − − j=2 − +1holds. ∏ 2 holds. The conditions (8) and (9) for every s , s Gk hold, so they hold for their product that is equivalent The conditions (8), (9) for every s2, s G hold∈ so it holds for their product that is equivalent to conditions which hold for every∈ commutator.k to conditions hold for every commutator.
Definition 3. We define a subdirect product of group Gk 1 with itself by equipping it with condition (8) and Definition 3. We define a subdirect product of group −Gk 1 with itself by equipping it with (9) of index parity on all of k 1 levels. − condition (8) and (9) of index parity− on all of k 1 levels. − CorollaryCorollary 24. The 9. The subdirect subdirect product productGGkk 11 Gkk 11isis defined defined by k by k2 outer2 outer relations relations on level onsubgroups. level The 2k k 2 −−2k k 2 −− − − subgroups.order The of G orderk 1 ofGkG1kis12−G−k .1 is 2 − − . − − − − G G (k 2) Proof. WeProof. specifyWe specify a subdirect a subdirect product product for the for thegroup groupk G1k 1 kGk1 1byby using using (k 2) conditionsconditions for the − − − − − for the subgroup levels. Each Gk 1 has even index on k 2-th level, it implies that− its relation subgroup levels. Each Gk −1 has even index on k 2−-th level; it implies that its relation for l = k 1 for l = k 1 holds automatically.− This occurs because− of the conditions of parity for the index − holds− automatically. This occurs because of the conditions of parity for the index of the last level is of the last level is characteristic of each of the multipliers Gk 1. Therefore It is not an essential characteristic of each of the multipliers Gk 1. Therefore, It− is not an essential condition for determining − conditiona subdirect for determining product. a subdirect product. Thus, to specify a subdirect product in the group Gk 1 Gk 1, there are obvious only Thus, to specify a subdirect product in the group− Gk 1 − Gk 1, one need only k 2 outer k 2 outer conditions on subgroups of levels. Any of such conditions− − reduces the order− of conditions on subgroups of levels. Any of such conditions reduces the orderk of1G G by two − 2 − k2 1 k 1 G G k 1 G 2 − × − k 1 k 1 in 2 times. Hence, taking into account that the order2 − of 2 k 1 is − , the order − × times.− Hence, taking into account that the order of Gk 1 is 2 − , we− can conclude that the2 order − 2k 1 2 k −1 2 of Gk 1 Gk 1 as a subgroup of Gk 1 Gk 1 the following: Gk 1 Gk 1 = 22 − −2 : k 2 − of Gk −1 Gk 1 as a subgroup of−Gk×1 G−k 1 is the following:| G−k 1 G−k |1 = 2 − : 2 − = k 2 2k 4− k 2− 2k k 2 − × − | − − | 2 − = 22k −4 :k 2 2− =2k 2 k −2 − . Thus, we use k 2 additional conditions on level( subgroup) 2 − : 2 − = 2 − − . Thus, we use k 2 additional− conditions on level subgroup to define the to define the subdirect product Gk 1 Gk 1,− which contain G′k as a proper subgroup of Gk. subdirect product Gk 1 Gk −1, which− contain G0k as a proper subgroup of Gk, because according to Because according to the conditions,− − which are realized in the commutator of G , (9) and (8) the conditions, which are realized in the commutator of G ,(9) and (8) indexes of′k levels are even. indexes of levels are even. 0k Corollary 10. A commutator G is embedded as a normal subgroup in G G . Corollary 25. A commutator G′k is0k embedded as a normal subgroup ink G1 k 1k1Gk 1. − − − −
Proof. AProof. proofA of proof injective of injective embedding embeddingG′k intoG0kGintok 1 GkG1k 1Gimmediatelyk 1 immediately follows follows from from last last item item of − − − − of proofproof of Corollary of Corollary 24.9. The The minimality minimality of ofG0kGas′k aas normal a normal subgroup subgroup of Gk and of G injectivek and embedding injective G0k embeddingintoGG′kk 1intoGGk k1 immediately1 Gk 1 immediately entails that entailsG0k / Gk that1 GGk′k 1▹. Gk 1 Gk 1. − − − − − − − − G G = G G TheoremTheorem 26. A 5. commutatorA commutator of subgroupk has form of G has′k form Gk 1 = G k 1, whereG , where the subdirect the subdirect product product is k −20k k 2k −1 k 1 is defined by relations (8) and (9). The order of G′k is 2 − − .− − 2k k 2 defined by relations (8) and (9). The order of G0k (the commutator subgroup of Syl2 A2k ) is 2 − − .
Proof. Since according to Statement 2 (g1, g2) as elements of G′k also satisfy relations (8) and Proof. Since according to Statement2 (g1, g2) as elements of G0k also satisfy relations (8) and (9), which (9), which define the subdirect product Gk 1 Gk 1. Also condition g1g2 B′k 1 gives parity define the subdirect product Gk 1 G−k 1. − ∈ − of permutation which defined by (g1,− g2) because− B′k 1 contains only element with even index Also g1g2 B0k 1 implies the parity of permutation− defined by (g1, g2), because B0k 1 contains of level [19]. The group∈ G−′k has 2 disjoint domains of transitivity so G′k has the structure− only an element with even index of level [15]. The group G has two disjoint domains of transitivity of a subdirect product of Gk 1 which acts on this domains0 transitively.k Thus, all elements − G so G0k has the structure of a subdirect product of Gk 1 which acts on thisG domainsG transitively. Thus, of ′k satisfy the conditions (8), (9) which define subdirect− product k 1 k 1. Hence G − − G G G′k < Gallk 1 elements Gk 1 ofbut 0Gk ′satisfyk can be the equipped conditions by (8 some) and other (9) which relations, define therefore, subdirect theproduct presencek 1 of k 1. − − − − isomorphismHence hasG0k not< G yetk 1 beenGk proved.1 but G0k Forcan proving be equipped revers by inclusion some other we relations; have to showtherefore, that the every presence − − 1 1 elementof from isomorphismGk 1 G hask 1 notcan yet be been expressed proven. as For word provinga− b− reversab, where inclusiona, b weG havek. Therefore, to show that it every − − 1 1 ∈ suffices toelement show from the reverseGk 1 inclusion.Gk 1 can be For expressed this goal as we some use word that aG−′kb− Mathematics 2020, 8, 472 15 of 19 Definition 3. We define a subdirect product of group Gk 1 with itself by equipping it with − condition (8) and (9) of index parity on all of k 1 levels. − suffices to show the reverse inclusion.Corollary For this goal 24. weThe use subdirect the fact that productG0k Since h satisfies the relations (8) and (9) for all 0 l k then h Gk 1 Gk 1. ≤ ≤ ∈ − − On the (k 1)-th level, we choose the generator τ which was defined in [15] as τ = τ τ k 1 . k 1, 1 k 1, 2 − − −k 2 − Recall that it was shown in [15] how to express any τ using τ, τ k 2 , τ k 2 , where i, j < 2 , in form ij i,2 − j,2 − − 1 1 of a product of commutators τij = τi,2k 2 τj,2k 2 = (α− τ− k 2 αiτ k 2 ). − − i 1,2 − j,2 − 1 1 Here τ k 2 was expressed as the commutator τ k 2 = α− τ− k 2 αiτ k 2 . i,2 − i,2 − i 1,2 − 1,2 − Thus, we express all tuples of elements satisfying to relations (8) and (9) by using only commutators of Gk. 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, k 2 [k 1] to obtain a tuple of v.p. with 2m active v.p. on X − of v11X − , we will construct the product for τij k 2 for varying i, j < 2 − . 4. Minimal generating set For the construction of minimal generating set we used the representation of elements of group Gk by portraits of automorphisms at restricted binary tree AutXk. For convenience we will identify [k] elements of Gk with its faithful representation by portraits of automorphisms from AutX . We denote by A a set of all functions a , such, that [ε, . . . , ε, a , ε, . . .] [A] . Recall that, |l l l ∈ l according to [21], l-coordinate subgroup U < G is the following subgroup. k 2 2 − (g l 1 g l 1 , ..., g l g l ). Therefore the product of form φ(gk 1,jg ) = Definition 1. For anl2 − arbitrarry+1 lσ2(2 −k+1) N wel(2 call) lσ a2(2k) coordinate subgroup U < G a subgroup, lσ(j) ∈ − j=1 − which is determined byk k1 -coordinate sets [U]l, l , if this subgroup consists of all Kaloujnine’s 2 − N ∏ 2 ∈ 2 tableaux a I for which [a]φl (gk[U1],jl.) = e, because of gk 1,j = e. Thus, characteristic equation (9) of k 1 level ∈ k 2 ∈ − − − j=2 − +1 l We denote by Gholds.k(l∏) a level subgroup of Gk, which consists of the tuples of v.p. from X , l < k 1 of any α Gk. We denote as Gk(k 1) 2such subgroup of Gk that is generated − ∈The conditionsk 1 (8), (9) for every− s , s Gk hold so it holds for their product that is equivalent by v.p., which are located on X − and isomorphic to W∈k 1. Note that Gk(l) is in bijective to conditions hold for every commutator.− correspondence (and isomorphism) with l-coordinate subgroup [U]l [21]. Mathematics 2020, 8, 472 l 16 of 19 For any v.p. gli Definitionin vli of X we 3. setWe in define correspondence a subdirect with productgli the of permutation group Gk 1φwith(gli) itselfS2 by equipping it with − ∈ the following rule: condition (8) and (9) of index parity on all of k 1 levels. − Thus, all vertex labelings of automorphisms,Corollary which appear 24. The in the subdirect representation product ofGGkk 11 Gkk 11 is defined by k 2 outer relations on level (1, 2), if gli =−e,k − [k] φ(gli) = ̸ −2 k 2 − − (7) by portraits as the subgroup of AutX , are alsosubgroups. in the representation The order of ofGGk0k.e,1 Gifk g1 is=2e.− − . { − − li Since there are faithful representations of Gk 1 Gk 1 and G0k by portraits of automorphisms from [k] Define a homomorphicProof.− We map− specify from G ak subdirect(l) onto S product2 with the for kernel the group consistingGk 1 of G allk products1 by using of (k 2) conditions AutX , which coincide with each other, subgroup G0k of Gk 1 Gk 1 G0k is equal to Gk 1 Gk 1 − − − even number of transpositionsfor the subgroup that− belongs levels.− ' toEachGk(Gl)k. For1 has instance, even− index the− element on k 2(12)(34)-th level,of itG impliesk(2) that its relation (i.e., Gk 1 Gk 1 = G0k). − − − − belongs to kerφ. Hence,for l =φk(gli)1 holdsS2. automatically. This occurs because of the conditions of parity for the index − ∈ G The archived results are confirmed by algebraicof the system last level GAP is calculations. characteristicl For ofinstance, each ofSyl the2 A multipliers8 = k 1. Therefore It is not an essential 6 23 2 Definition23 3 2 2. We define the subgroup of -th level as a subgroup| generated| by all− possible vertex 2 = 2 and (SylA 3 ) = 2 = 8. Thecondition order of G foris determining 4, the number a ofsubdirect additional product. relations in − | 2 0permutation| − − of this level. 2 the subdirect product is k 2 = 3 2 = 1. We haveThus, the to same specify result a(4 subdirect4) : 21 = product8, whichin confirms the group Gk 1 Gk 1, there are obvious only − − · − − Theorem5. Statement 2. In Gk k′,2 theouter following conditionsk equalities on subgroups are true: of levels. Any of such conditions reduces the order of − 2k 1 2 Gk 1 Gk 1 in 2 times. Hence, taking into account that the order of Gk 1 is 2 − − , the order − − l − × 2 k 1 2 Example 1. Set k = 4 then (SylA )0 = (G )0 = 1024, G3 = 64, since k 2 = 2, so according to our 2 − 2 | 16 | | 4 of| Gk 1 | Gk| 1 as a subgroup− of Gk 1 Gk 1 the following: Gk 1 Gk 1 = 2 − : − k 2 − 2 φ(glj) = e, 0 l <− k ×1. − | − (8)− | theorem above order of Syl2 A16 Syl2 A16 is defined by 2 −k = 2 relations, andk by≤ this reason− is equal to 2k 2 = 22 4 l=1: 2k 2 = 22 k 2. Thus, we use k 2 additional conditions on level( subgroup) (64 64) : 4 = 1024. Thus, orders are coincides. − − ∏ − − − − · to define the subdirect product Gk 1 Gk 1, which contain G′k as a proper subgroup of Gk. For the case i = k 1, the following condition holds:− − 25 G Example 2. The true order of (Syl2 A32)0 is 33554432Because− = 2 according, k = 5. A to number the conditions, of additional which relations are which realized in the commutator of ′k, (9) and (8) indexes of levels are even. 14 14 define the subdirect product is k 2 = 3. Thus, according to Theorem5, (Syl2 A16 Syl2 A16)0 = 2 2 : 5 2 28 5 2 25 − 2k 2 | 2k 1 | 2 − = 2 : 2 − = 2 . − − Corollary 25. A commutator G′k is embedded as a normal subgroup in Gk 1 Gk 1. φ(gk 1j) = φ(gk 1j) = e. (9)− − − − j=1 k 2 According to calculations in GAP we have:Proof.SylA2 A proof7 ofSyl injective2 A6 Dj=2 embedding4.− Therefore,+1 G′ itsk into derivedGk 1 Gk 1 immediately follows from last item ∏' ' ∏ − − subgroup (Syl2 A7)0 (Syl2 A6)0 (D4)0 = Cof2. proof of Corollary 24. The minimality of G′k as a normal subgroup of Gk and injective ' Thus,' G′ has k new conditions on a combination of level subgroup elements, except for the The following structural law fork Syllowsembedding 2-subgroupsG′ isk into typical.Gk The1 structuresGk 1 immediately of Syl2 An entailsand that G′k ▹ Gk 1 Gk 1. condition of last level parity from the original− group.− − − Syl2 Ak are the same if all n and k have the same multiple of two as the multiplier in decomposition on Theorem 26. A commutator of Gk has form G′k = Gk 1 Gk 1, where the subdirect product n! and k! Thus, Syl2 A2k Syl2 A2k+1. −k − 'Proof. Note that the condition (8) is compatible with that were foundedG by22 R.k Guralnik2 in [27], is defined by relations (8) and (9). The order of ′k is −k −2 . − (i) Example 3. Syl2 A7 Sylbecause2 A6 asD4 it, Syl was2 A proved10 Syl by2 A author11 Syl [19]2S8Gk (1D4 BkD42)oC2.k Syl1,2 whereA12 Bk 2 C2 . ' ' Proof.' Since' according' to− Statement≃ × − 2W(g−1, g2) as' elements− ≃ i of=1≀ G′k also satisfy relations (8) and Syl2S8 Syl2S4, by the same reasons that from the proof of Corollary9 its2 commutator subgroup is decomposed According to Property(9), which 1, G define′k G thek, sosubdirect it is enough product to proveGk 1 theGk statement1. Also condition for the elementsg1g2 B′k 1 gives parity as (Syl2 A12)0 (Syl2S8)0 (Syl2 2S4)0. ≤ − − ∈ − ' of× Gk. Such elements,of permutation as it was described which defined above, can by ( beg1, presented g2) because in theB′k form1 containss = (sl1 only, ..., s elementl2l )σ, with even index l − where σ Gl 1 andofs levelli are [19]. states The of s groupGkGin′k vhasli, i 2 disjoint2 . For domains convenience of transitivity we will make so G the′k has the structure Lemma 7. In Gk00 the following equalities∈ are− true: ∈ ≤ transition from theof tuple a subdirect(sl1, ..., s productl2l ) to the of G tuplek 1 (whichgl1, ..., acts gl2l ). on Note this that domains there transitively. is the trivial Thus, all elements 2 − l 2 vertexl 1 permutationl 1 ofgl lj2G=′k esatisfyin the theproductl conditions of the states (8), (9)slj whichslj. define subdirect product Gk 1 Gk 1. Hence 2 − 2 − 2 − +2 − 2 · − − G G′k [k 2] [k 2] In other terms, the subgroup Gk00 has an even index of any level of v11X − and of v12X − . The order of 2k 3k+1 Gk00 is equal to 2 − . Proof. As a result of derivation of G , elements of G (1) are trivial. Due the fact that G k0 k00 0k ' Gk 1 Gk 1, we can derivate G0k by commponents. The commutator of Gk 1 is already investigated in − − 2 − Theorem5. As Gk 1 = G0k 1 by Corollary7, it is more convenient to present a characteristic equalities − − 2 2 in the second commutator G00k G0k 1 G0k 1 as equations in Gk 1 Gk 1. As shown above, for 2 ' − − − − 2 l < k 1, in Gk 1 the following equalities are true: ≤ − − l 1 l 1 l 1 l 1 l 1 l 1 2 − 2 − 2 − 2 − 2 − 2 − 2 ∏ ϕ(gljglσ(j)) = ∏ ϕ(glj) ∏ ϕ(glσ(j)) = ∏ ϕ(glj) ∏ ϕ(gli) = ∏ ϕ(glj) = e (13) j=1 j=1 j=1 j=1 j=1 j=1 k 2 2 − (g l 1 g l 1 , ..., g l g l ) φ(g g ) = l2 − +1 lσ2(2 − +1) l(2 ) lσ2(2 ) . Therefore the product of form k 1,j lσ(j) j=1 − k 1 2 − ∏ 2 2 φ(gk 1,j) = e, because of gk 1,j = e. Thus, characteristic equation (9) of k 1 level k 2 − − − j=2 − +1 Mathematics 2020, 8, 472 holds.∏ 17 of 19 The conditions (8), (9) for every s2, s G hold so it holds for their product that is equivalent ∈ k to conditions hold for every commutator. l 2 l 1 l 1 l 2 l 2 − 2 − 2 − +2 − 2 ϕ(g ) = ϕ(g ) = ϕ(g ) = ϕ(g ). (14) ∏ Definitionlj ∏ 3. Welj define∏ a subdirectlj product∏ of grouplj Gk 1 with itself by equipping it with j=1 l 2 l 1 l 1 l 2 − conditionj=2 − (8)+1 and (9) ofj= index2 − +1 parity onj= all2 − of+2k− +11 levels. − The equality (14) holdsCorollary since it is valid 24. inThe the subdirect initial group productG0k GGkk 11 Gkk 11. Theis defined equalities by k 2 outer relations on level ' −−2k k 2 −− − subgroups. The order of Gk 1 Gk 1 is 2 − − . l 1 l 2 l 2 − +2 − − 2 − Proof. We specify∏ ϕ( aglj subdirect) = ∏ productϕ(g forlj) the group Gk 1 Gk 1 by using (k 2) conditions j=2l 1+1 j=2l 1+2l 2+1 − − − for the subgroup− levels. Each− Gk− 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 hold for elements of second− group G0k 1, since the elements of the original group are endowed of the last level is characteristic− of each of the multipliers Gk 1. Therefore It is not an essential with these conditions. − 2 condition for determining a subdirect product. In (G0k) any element g of G0k(l) satisfies the equality (14). Moreover, g satisfies the previous Thus, to specify a subdirect product in the group Gk 1 Gk 1, there are obvious only 2 − − conditions (11) because of (Gk 1(l)) = G0k 1(l). k 2 outer− conditions− on subgroups of levels. Any of such conditions reduces the order of 2 k 1 The similar conditions− appear in (G0k 1(k 2)) after squaring of G0k. Thus, taking into account 2 − 2 Gk 1 Gk 1 in 2 times.− − Hence, taking into2 account that the order of Gk 1 is 2 − , the order the characteristic equations− of× G − (l), the subgroup (G (k 2)) satisfies the equality: − 2 0k 1 0k 1 2k 1 2 of Gk 1 −Gk 1 as a subgroup of− Gk−1 Gk 1 the following: Gk 1 Gk 1 = 2 − − : − − − × − | − − | k 2 2k 4 k 2 2k k 2 ( ) k 3 2 − =k 2 2 − : 2 − =k 22 −k 3− . Thus, wek use1 k 2 additional conditions on level subgroup 2 − 2 − 2 − +2 − 2 − − to define the subdirect product Gk 1 Gk 1, which contain G′k as a proper subgroup of Gk. ∏ ϕ(glj) = ∏ ϕ(glj) = e, ∏ ϕ(glj)− = −∏ ϕ(glj) = e. (15) k 3 k 2 k 1 k 2 G j=1 Becausej=2 − + according1 to thej=2 − conditions,+1 whichj=2 − + are2 − realized+1 in the commutator of ′k, (9) and (8) indexes of levels are even. Taking into account the structure G0k Gk 1 Gk 1, we obtain after the derivation G00k Corollary 25. A commutator' − G′−k is embedded as a normal subgroup in' Gk 1 Gk 1. (Gk 2 Gk 2) (Gk 2 Gk 2). With respect to conditions (8) and (9) in the subdirect product, we − − − − − − k k have that the order of G is 22 k 2 : 22k 3 = 22 3k+1 because on each level 2 l < k, the order Proof.00k A proof− − of injective− − embedding G′k into Gk 1 Gk 1 immediately follows from last item − ≤ − of level subgroup G00kof(l) proofis 4 times of Corollary smaller than 24. the The order minimality of G0k(l) of. OnG′k theas first a normal level, one subgroup new of Gk and injective condition arises that reducesembedding the orderG′k into of GG0kk(1)1by G 2 times.k 1 immediately In total, we entails have 2( thatk 2)G +′k1▹= G2kk 1 3 Gk 1. − − − −− − new conditions for comparing with G0k. Theorem 26. A commutator of Gk has form G′k = Gk 1 Gk 1, where the subdirect product −2k k 2 − Corollary 11. Any minimalis defined generating by relations set of Syl (8)A k and, k > (9).2 consists The order of 2k of3 Gelements.′k is 2 − − . 02 2 − Proof. The proof is basedProof. onSince two facts according about toG02 StatementG00 G02 = 2 (Gg1,. g2 More) aselements precisely itof isG based′k also on satisfy relations (8) and k k ' k 0k (9), which define the subdirect2 product2k 3 Gk 1 Gk 1. Also condition g1g2 B′k 1 gives parity Corollary7 and on a calculating of the index G0 : G0 G00 = 2 − . − − ∈ − of permutation which definedk k by (g1, g2) because B′k 1 contains only element with even index 2 2k 3 − To justify that theof index level [19].G0 : G0 TheG00 group= 2 −G,′ wek has take 2 into disjoint consideration domains the of transitivity orders of these so G′k has the structure k k subgroups from Theoremof a5 subdirectand Lemma product7. Corollary of G7k tell1 which us that acts the subgroup on this domainsG2 is equal transitively. to the Thus, all elements − k of G′k satisfy the conditions (8),2 (9) which define subdirect product Gk 1 Gk 1. Hence subgroup Gk0 , then the Frattiny subgroup Φ(Gk0 ) = Gk00 = Gk0 . According to Corollary7 the subgroup − − G′k < Gk 1 Gk 1 but G′k can be equipped by some2 other relations, therefore, the presence of G2 is equal to the subgroup G ,− then the− Frattiny subgroup Φ(G ) = G = G . Further, for finding k isomorphismk0 has not yet been proved. Fork0 provingk00 k0 revers inclusion we have to show that every 1 1 00 the Frattiny factor, whichelement is an elementary from Gk 1 abelian Gk 2-group,1 can be it isexpressed enough sufficient as word toa− calculateb− ab, whereG0 : Gk a, b Gk. Therefore, it − − ∈ because of Φ(Syl20 A2k )suffices = Syl200 to(A2 showk ). Due the to reverse Lemma inclusion.7, we have ForG00 thisk 1 goalG0k we2 useG0k that2, henceG ′k < the G k 1 Gk 1. As it was k − ' − − − − 2k 3k+1 2 2 order of G00k 1 is equalshown to 2 in− [19]. that Taking the into order account of Gk thatis 2 G−00k .is normal subgroup of G0k, we − 2k 3 compute the order of FrattinyAs it was quotient shown is 2 above,− . Thus,G′k has accordingk new conditionsto Frattiny relativelytheorem, a to minimalGk. Each condition is stated generating set of Syl onA k someconsists level-subgroup. of 2k 3 elements. Each It ofis well these known conditions [28], the reduces ordersan of irreducible order of the corresponding level 02 2 − generating sets for p-group are equal to each other. In case k = 2 the Syl A K , therefore the commutator subgroup is trivial. 2 4 ' 4 2 Example 4. The size of (G400) is 32. The size of the direct product (G30 ) is 64, but, due to relation on second 2 level of Gk00, the direct product (G30 ) transforms into the subdirect product G30 G30 that has two times less feasible combination on X2. The number of additional relations in the subdirect product is k 3 = 4 3 = 1. − − Thus, the order of product is reduced by 21 times. Mathematics 2020, 8, 472 18 of 19 Example 5. The commutator subgroup of Syl (A ) consists of elements: e, (13)(24)(57)(68), 20 8 { (12)(34), (14)(23)(57)(68), (56)(78), (13)(24)(58)(67), (12)(34)(56)(78), (14)(23)(58)(67) . The } commutator Syl (A ) C3 is an elementary abelian 2-group of order 8. This fact confirms our formula 20 8 ' 2 d(G ) = 2k 3, because k = 3 and d(G ) = 2k 3 = 3. A minimal generating set of Syl (A ) consists of k − k − 20 8 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 Syl (A ) consists of five (that is 2 4 3) generators: 20 16 · − (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) × (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 Syl (A ) consists of seven (that is 2 5 3) generators: 20 32 · − (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 G4 can be presented in form of wreath recursion: a1 = (e, e)σ, b2 = (a1, e) , a3 = (b2, e) , b4 = (b3, b3) , where σ = (1, 2). The minimal generating set for G04 can be presented in form of wreath recursion: a2 = (σ, σ), a3 = (e, a2), a4 = (a3, a3) , b3 = (e, b2), b4 = (b3, b3). where σ, a3, a4 are generators of the first multiplier G3 and σ, b3, b4 are generators of the second. 5. Conclusions The size of minimal generating set for commutator of Sylow 2-subgroup of alternating group A2k was proven to be equal to 2k 3, where k > 2. − A new approach to presentation of Sylow 2-subgroups of alternating group A2k was applied. As a result, the short proof of a fact that commutator width of Sylow 2-subgroups of the alternating group A2k (k > 2), permutation group S2k and Sylow p-subgroups of Syl2 Apk (Syl2Spk ) are equal to 1 was obtained. Commutator widths of permutational wreath products B C were investigated. o n We constructed the minimal generating set of the commutator subgroup of the Sylow 2-subgroup of the alternating group. The inclusion problem [18] for Syl2 A2k and its subgroups as (Syl2 A2k )0 and (Syl2 A2k )00 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. Funding: This research was funded by Interregional Academy of Personnel Management. Conflicts of Interest: The author declares no conflict of interest. References 1. Muranov, A. Finitely generated infinite simple groups of infinite commutator width. Inter. J. Algebra Comput. 2007, 17, 607–659. [CrossRef] 2. Gordeev, N.L.; Kunyavskii, B.E.; Plotkin, E.B. Geometry of word equations in simple algebraic groups over special fields. Russ. Math. Surv. 2018, 73, 753–796. [CrossRef] 3. Bartholdi, L.; Groth, T.; Lysenok, I. Commutator width in the first Grigorchuk group. arXiv 2017, arXiv:1710.05706. Mathematics 2020, 8, 472 19 of 19 4. Calegari, D. MSJ Memoirs; Mathematical Society of Japan: Tokyo, Japan, 2009; Volume 20, MR 2527432, p. 209. 5. 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