Central Automorphisms and Inner Automorphisms in Finitely Generated Groups

Abstract

Let G be a group and Aut_c(G) be the group of all central automorphisms of G. We know that in a finite p-group G, Aut_c(G) = Inn(G) if and only if Z(G) = G′ and Z(G) is cyclic. But we shown that we cannot extend this result for infinite groups. In fact, there exist finitely generated nilpotent groups of class 2 in which G′ =Z(G) is infinite cyclic and Inn(G) < C* = Aut_c(G). In this article, we characterize all finitely generated groups G for which the equality Aut_c(G) = Inn(G) holds.

Key Words:

• Central automorphism
• Finitely generated group
• Inner automorphism

2010 Mathematics Subject Classification:

• Primary: 20D45
• Secondary: 20E36

ACKNOWLEDGMENT

The author thanks the editors of Communications in Algebra and the referees who have patiently read and verified this article. The author also likes to acknowledge the support of Alzahra University.