Finite p-Groups All of Whose Proper Quotient Groups are Abelian or Inner-Abelian

Abstract

In this article, finite p-groups all of whose proper quotient groups are abelian or inner-abelian are classified. As a corollary, finite p-group all of whose proper quotient groups are abelian, and finite p-groups all of whose proper sections are abelian or inner-abelian are also classified.

Key Words:

• Extraspecial p-groups
• Inner abelian p-groups
• Metacyclic p-groups
• Minimal non-abelian p-groups
• Outer abelian p-groups
• p-groups of maximal class

2000 Mathematics Subject Classification:

• 20D15

ACKNOWLEDGMENTS

We cordially thank the referee for her(his) detailed and valuable comments, which help us to improve the whole article. Thanks also go to Professor Ben Brewster, who read carefully our manuscript and gave many helpful comments.

This work was supported by NSFC (No. 10671114), by NSF of Shanxi Province (No. 2008012001) and The Returned Abroad-student Found of Shanxi Province (No. [2007]13-56.

Notes

Communicated by A. Turull.