Minimal generating set and properties of commutator of Sylow subgroups of alternating and symmetric groups

Abstract

Given a permutational wreath product sequence of cyclic groups of order 2 we research a commutator width of such groups and some properties of its commutator subgroup. The paper presents a construction of commutator subgroup of Sylow 2-subgroups of symmetric and alternating groups. Also minimal generic sets of Sylow 2-subgroups of A2k were founded. Elements presentation of (Syl2A2k)', (Syl2S2k)' was investigated. We prove that the commutator width of an arbitrary element of a discrete wreath product of cyclic groups Cpi, pi ∈ N is 1.