Maximal abelian subgroups of the finite symmetric group

Document Type : Research Paper


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


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 $2$.


Main Subjects

[1] R. Bercov and L. Moser, On Abelian permutation groups, Canad. Math. Bull., 8 (1965) 627–630.
[2] J. M. Burns and B. Goldsmith, Maximal order abelian subgroups of symmetric groups, Bull. London Math. Soc.,
21 (1989) 70–72.
[3] J. D. Dixon, Maximal abelian subgroups of the symmetric groups, Canad. J. Math., 23 (1971) 426–438.
[4] L. A. Kaluˇznin and M. H. Klin, Certain maximal subgroups of symmetric and alternating groups (Russian), Mat.
Sb. (N.S.), 87 (1972) 91–121.
[5] J. Konieczny, Centralizers in the semigroup of injective transformations on an infinite set, Bull. Aust. Math. Soc.,
82 (2010) 305–321. 
[6] J. Konieczny, Second centralizers in the semigroup of injective transformations, Asian-Eur. J. Math., 3 (2020) 11
[7] M. W. Liebeck, C. E. Praeger and J. Saxl, A classification of the maximal subgroups of the finite alternating and
symmetric groups, J. Algebra, 111 (1987) 365–383.
[8] B. Newton and B. Benesh, A classification of certain maximal subgroups of symmetric groups, J. Algebra, 304
(2006) 1108–1113.
[9] B. K. Sahoo and B. Sahu, Maximal elementary abelian subgroups of the symmetric group, J. Appl. Algebra Discrete
Struct., 4 (2006) 47–56.
[10] M. Suzuki, Group Theory I, Springer-Verlag, New York, 1982.