On the Rarity of Quasinormal Subgroups

For each prime p and positive integer n, Berger and Gross have defined a finite p-group G = HX, where H is a core-free quasinormal subgroup of exponent p^n-1 and X is a cyclic subgroup of order pⁿ. These groups are universal in the sense that any other finite p-group, with a similar factorisation into subgroups with the same properties, embeds in G. In our search for quasinormal subgroups of finite p-groups, we have discovered that these groups G have remarkably few of them. Indeed when p is odd, those lying in H can have exponent only p, p^n-2 or p^n-1. Those of exponent p are nested and they all lie in each of those of exponent p^n-2 and p^n-1.