Isomorphism between automorphism groups of finitely generated groups

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 technical lemma and, as a consequence, obtain a short and easy proof of this result of Azhdari and Malayeri. Subsequently, we also obtain short proofs of some other existing and some new related results.