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

Base (minimal generating set) of the Sylow 2-subgroup of (formula presented) 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 (formula presented) 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 2ⁿ⁻¹ diagonal bases and (formula presented) bases at all. Recursive construction of Cayley graphs of P_n(2) on diagonal bases (n ≥ 2) is proposed.