COMMUTATOR WIDTH IN THE WREATH PRODUCT, COMMUTATOR SUBGROUP OF SYLOW 2-SUBGROUPS OF ALTERNATING GROUP AND SYMMETRIC GROUPS

摘要

The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroups of alternating group is found. The structure of commutator subgroup of Sylow 2-subgroups of the alternating group A2k is investigated. It is shown that (Syl2A2k ) 2 = Syl0 2A2k , k > 2. It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups Cpi , pi ∈ N equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. This paper presents upper bounds of the commutator width (cw(G)) [1] of a wreath product of groups. A recursive presentation of Sylows 2-subgroups Syl2(A2k ) of A2k is introduced. As a result the short proof that the commutator width of Sylow 2-subgroups of alternating group A2k , permutation group S2k and Sylow p-subgroups of Syl2Apk (Syl2Spk ) are equal to 1 is obtained. A commutator width of permutational wreath product B o Cn is investigated. An upper bound of the commutator width of permutational wreath product B o Cn for an arbitrary group B is found.