The girth of Cayley graphs over Sylow 2-subgroups of the symmetric groups S_2n with diagonal bases

A diagonal base of a Sylow 2-subgroup Pn(2) of symmetric group S2n is a minimal generating set of this subgroup consisting of elements with only one nonzero coordinate in the polynomial representation. For different diagonal bases, Cayley graphs over Pn(2) may have different girths (i.e. minimal lengths of cycles). In this paper, all possible values of girths of Cayley graphs over Pn(2) with diagonal bases are calculated. A criterion for whenever such Cayley graph has girth equal to 4 is presented.