On equality of certain automorphism groups of finite groups

ABSTRACT

Let G be a finite group and L(G) denotes the absolute center of G. An automorphism α of G is called an autocentral automorphism if x⁻¹α(x)∈L(G) for each x∈G. An automorphism α of G is called a central automorphism if x⁻¹α(x)∈Z(G) for each x∈G. An automorphism α of G is called an IA-automorphism if $x^{-1}\alpha \left(x\right)\in G^{\prime }$ for each x∈G. In this paper, we find necessary and sufficient conditions on G such that every autocentral automorphism is inner. Also we characterize finite non-Abelian p-groups for which every central automorphism is IA-automorphism.

KEYWORDS:

• Absolute centre
• autocentral automorphisms
• central automorphisms
• IA-automorphisms

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D45
• 20D15

Acknowledgements

The author is grateful to the referee for his/her helpful comments.