Maximal abelian subgroups of the finite symmetric group

Document Type : Research Paper

Author

Department of Mathematics, University of Mary Washington, Fredericksburg, Virginia, USA

Abstract

Let $G$ be a group. For an element $a\in G$, denote by $C^2(a)$ 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 $C^2(a)\subseteq M$ 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 $\bigcup_{a\in A}C^2(a)$ 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 $S_n$ of a-rank $1$ and describe a class of maximal abelian subgroups of $S_n$ of a-rank at most $2$.

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References

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International Journal of Group Theory
Volume 10, Issue 3 - Serial Number 3
September 2021
Pages 103-124

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History

  • Receive Date: 07 March 2020
  • Revise Date: 24 April 2020
  • Accept Date: 25 April 2020
  • Published Online: 01 September 2021

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