For each prime and positive integer , Berger and Gross have defined a finite -group , where is a core-free quasinormal subgroup of exponent and is a cyclic subgroup of order . These groups are universal in the sense that any other finite -group, with a similar factorisation into subgroups with the same properties, embeds in . In our search for quasinormal subgroups of finite -groups, we have discovered that these groups have remarkably few of them. Indeed when is odd, those lying in can have exponent only , or . Those of exponent are nested and they all lie in each of those of exponent and .
Cite this article
John Cossey, Stewart Stonehewer, On the Rarity of Quasinormal Subgroups. Rend. Sem. Mat. Univ. Padova 125 (2011), pp. 81–105