The commutator and centralizer description of Sylow 2-subgroups of alternating and symmetric groups

الأصل · EN

Given a permutational wreath product sequence of cyclic groups of prime order we research a commutator width of such groups and some properties of its commutator subgroup. Commutator width of Sylow 2-subgroups of alternating group A₂ᵏ, permutation group S₂ₖ and Cₚ B were founded. The result of research was extended on subgroups (Syl₂ A₂ₖ)', p>2. 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 (Syl₂ A₂ₖ)', (Syl₂ S₂ₖ)' was investigated. We prove that the commutator width Mur of an arbitrary element of a discrete wreath product of cyclic groups Cₚ is 1. Key words: wreath product of group, commutator width of p-Sylow subgroups, commutator subgroup, centralizer subgroup, semidirect product.