This article is cited in 5 papers

RESEARCH ARTICLE

The action of Sylow 2-subgroups of symmetric groups on the set of bases and the problem of isomorphism of their Cayley graphs

Bartłomiej Pawlik

Institute of Mathematics, Silesian University of Technology, ul. Kaszubska 23, 44-100 Gliwice, Poland

Abstract: Base (minimal generating set) of the Sylow 2-subgroup of $S_{2^n}$ is called diagonal if every element of this set acts non-trivially only on one coordinate, and different elements act on different coordinates. The Sylow 2-subgroup $P_n(2)$ of $S_{2^n}$ acts by conjugation on the set of all bases. In presented paper the stabilizer of the set of all diagonal bases in $S_n(2)$ is characterized and the orbits of the action are determined. It is shown that every orbit contains exactly $2^{n-1}$ diagonal bases and $2^{2^n-2n}$ bases at all. Recursive construction of Cayley graphs of $P_n(2)$ on diagonal bases ($n\geq2$) is proposed.

Keywords: Sylow $p$-subgroup, group base, wreath product of groups, Cayley graphs.

MSC: 20B35, 20D20, 20E22, 05C25

Received: 10.04.2016
Revised: 30.05.2016

Language: English

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