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 ( S y l 2 A 2 k ) 2 = S y l 2 ′ 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 ∈ N equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. Upper bounds for the commutator width ( c w ( G ) ) of a wreath product of groups are presented in this paper. A presentation in form of wreath recursion of Sylow 2-subgroups S y l 2 ( A 2 k ) 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 S y l 2 A p k as well as S y l 2 S p k ) are equal to 1 was obtained. A commutator width of permutational wreath product B ≀ C n is investigated. An upper bound of the commutator width of permutational wreath product B ≀ 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.
Highlights
The object of our study is the commutatorwidth [1] of Sylow 2-subgroups of alternating group A2k
In related work [3], it was established that the commutator width of the first Grigorchuk group is 2
It is 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 ∈ G 0
Summary
The object of our study is the commutatorwidth [1] of Sylow 2-subgroups of alternating group A2k. As it is well known, the first example of a group G with commutator width greater than 1. We obtain an upper bound for commutator width of wreath product Cn o B, where Cn is cyclic group of order n, in terms of the commutator width cw( B) of passive group B. Elements of is proposed byofuscommutator-group as wreath recursion serves [8] andthe commutator width of it was studied. It has not been proven that commutator subgroup wreath products of nonabelian finite simple groups is bounded by an absolute constant [3,13]. We generalize passive of group of this wreath product to any o C not it was proven of that commutator subgroup of ≀ the.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have