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

Bartłomiej Tadeusz Pawlik

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

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References


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