Flatness Conditions on Finite p-Groups

Abstract

A group is flat if every finite conjugacy class is a coset of a (necessarily normal) subgroup. This paper is the first to consider the consequences of flatness condition for finite p-groups. The flatness condition is weaker than that of Camina pairs (as we shall prove in this paper) and a key aim of this paper is to prove a similar main result to that in the theory of such pairs. Consequences for orders, nilpotence classes and orders of centre of the groups are deduced.

Key Words:

• p-group
• Nilpotent
• Flat
• Camina pair
• Camina group

Mathematics Subject Classification:

• Primary 20D15
• Secondary 20C15

Acknowledgments

This paper is a modification of a part of the first author's Ph.D. thesis in the School of Informatics and Engineering, The Flinders University of South Australia.

The authors would like to thank John Humphreys of the University of Liverpool for kindly providing them with the example of a flat nilpotent group of class 3 and of order p ⁵, which was seminal to much of their work.

The authors also would like to thank the referee for all of the valuable suggestions (including the useful suggested citations), especially on the importance of discussing the relationship between flatness and Camina pairs of finite groups.

Notes

^#Communicated by A. Turull.

^‡On leave from The Department of Mathematics, Bandung Institute of Technology, Jl.Ganesha 10, Bandung 40132, Indonesia.