Automorphism groups of finitely generated groups that are isomorphic to group of inner automorphisms

Abstract

Let G be a finitely generated group and let C* denote the group of all central automorphisms of G fixing the center of G elementwise. Azhdari and Malayeri [J. Algebra Appl. 6 (2011), 1283–1290] gave necessary and sufficient conditions on G such that C* ≃ Inn(G). We prove a more general technical lemma and, as a consequence, obtain a short and easy proof of this result of Azhdari and Malayeri and generalize their result; and also generalize the main result of Nasrabadi and Farimani [Indag. Math. (N. S.) 26 (2015), 137–141] from finite autonilpotent p-groups of class 2 to arbitrary finitely generated nilpotent groups of class 2. Subsequently, we also obtain short proofs of some other existing and some new related results.