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Maximal abelian subgroups of the finite symmetric group | ||
| International Journal of Group Theory | ||
| مقاله 2، دوره 10، شماره 3، آذر 2021، صفحه 103-124 اصل مقاله (281.48 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2020.122036.1603 | ||
| نویسنده | ||
| Janusz Konieczny* | ||
| Department of Mathematics, University of Mary Washington, Fredericksburg, Virginia, USA | ||
| چکیده | ||
| 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 $2$. | ||
| کلیدواژه‌ها | ||
| Symmetric groups؛ maximal abelian subgroups؛ second centralizers؛ abelian rank | ||
| مراجع | ||
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[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 pp. [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. | ||
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