## Abstract

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 p^{n}. 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}.

Original language | English |
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Pages (from-to) | 81-105 |

Number of pages | 25 |

Journal | Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova |

Volume | 125 |

DOIs | |

Publication status | Published - 2011 |