Automorphisms of Finite Groups

Abstract

We study the classification of those finite groups G having a non-inner class preserving automorphism. Criteria for these automorphisms to be inner are established. Let G be a nilpotent-by-nilpotent group and S ∈ Sy l ₂(G). If S is abelian, generalized quaternion or S is dihedral, and in this case G is also metabelian, then Out _c (G) = 1. If S is generalized quaternion, 𝒵(S) ⊂ 𝒵(G) and S ₄ is not a homomorphic image of G, then Out _c (G) = 1. As a consequence, it follows that the normalizer problem of group rings has a positive answer for these groups.

Key Words:

• Automorphism
• Class preserving
• Group
• Conjugate

2000 Mathematics Subject Classification:

• 20E36
• 16U70

Acknowledgments

The first author would like to thank the Departamento de Matemática da Universidade Federal do Ceará for its warm hospitality, financial support and working conditions. Research was partially supported by CNPq-Brazil (Proc. 300652/95-0) CAPES-Brasil and FINEP-Brasil.

Notes

^#Communicated by M. Ferrero.