MAXIMAL ABELIAN SUBGROUPS OF THE FINITE SYMMETRIC GROUP

Let G be a group. For an element a ∈ G, denote by C(a) the second centralizer of a in G, which is the set of all elements b ∈ G such that bx = xb for every x ∈ G that commutes with a. Let M be any maximal abelian subgroup of G. Then C(a) ⊆ M for every a ∈ M. The abelian rank (a-rank) of M is the minimum cardinality of a set A ⊆ M such thata∈A(a) generates M. Denote by S the symmetric group of permutations on the set X = {1, …, n}. The aim of this paper is to determine the maximal abelian subgroups of S of a-rank 1 and describe a class of maximal abelian subgroups of S of a-rank at most 2.