Maximal abelian subgroups of the finite symmetric group

Abstract

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

Keywords

• ‎‎symmetric groups‎
• ‎maximal abelian subgroups‎
• ‎second centralizers‎
• ‎abelian rank