Higher Indicators for Some Groups and Their Doubles

Abstract

In this paper we explicitly determine all indicators for groups isomorphic to the semidirect product of two cyclic groups by an automorphism of prime order, as well as the generalized quaternion groups. We then compute the indicators for the Drinfel'd doubles of these groups. This first family of groups include the dihedral groups, the non-abelian groups of order pq, and the semidihedral groups. We find that the indicators are all integers, with negative integers being possible in the first family only under certain specific conditions.

Examples of non-$FSZ$ $p$-groups for primes greater than three

“…, are higher dimensional analogues of the integer parameter d appearing in [10]. The parameter d controlled the existence of negative indicators in the double of the groups under consideration in [10], in much the same way that Y p,j will dictate the existence of non-integer indicators here.…”

“…The parameter d controlled the existence of negative indicators in the double of the groups under consideration in [10], in much the same way that Y p,j will dictate the existence of non-integer indicators here. More generally, such objects naturally arise when considering the F SZ property for groups of the form A ⋊ C where A is abelian and C is cyclic, such as in [8,Example 4.4].…”

“…In full generality these indicators need not even be real numbers [9,Example 7.5] [7], but in the case of D(G) they are guaranteed to be so [8,Remark 2.8]. All of the first examples computed in the case of group doubles [4,10,11] yielded indicator values in Z. Since the higher indicators for G itself are classically known to be integers, this raised the question of whether or not the indicators for D(G) were always integers for arbitrary G.…”

The FSZ properties of sporadic simple groups

“…Proof. It is well-known that Ly contains a copy of G 2 (5) as a maximal subgroup, and that the order of Ly is not divisible by 5 7 . Therefore Ly and G 2 (5) have isomorphic Sylow 5-subgroups, and by Theorem 4.1 this Sylow subgroup is not F SZ 5 .…”

Some behaviors of FSZ groups under central products, central quotients, and regular wreath products

“…Several additional examples of F SZ + and non-F SZ groups can be found in [2,6,7,8,9,15]. It has been an open question as to whether or not there exists an F SZ group which is not F SZ + .…”