On Inner Automorphisms and Central Automorphisms of Nilpotent Group of Class 2

Abstract

Let G be a group and let Aut c(G) be the group of all central automorphisms of G. Let C* = C Aut c(G)(Z(G)) be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper, we prove that if G is a finitely generated nilpotent group of class 2, then C* ≃ Inn (G) if and only if Z(G) is cyclic or Z(G) ≃ Cm × ℤr where [Formula: see text] has exponent dividing m and r is torsion-free rank of Z(G). Also we prove that if G is a finitely generated group which is not torsion-free, then C* = Inn (G) if and only if G is nilpotent group of class 2 and Z(G) is cyclic or Z(G) ≃ Cm × ℤr where [Formula: see text] has exponent dividing m and r is torsion-free rank of Z(G). In both cases, we show G has a particularly simple form.

Central Automorphisms and Inner Automorphisms in Finitely Generated Groups

“…In [2] we characterized all finitely generated nilpotent groups of class 2 for which C * Inn G . Furthermore, we characterized all finitely generated nilpotent group for which C * = Inn G .…”

“…In [2,4] we characterized all finitely generated groups for which C * = Inn G . In [3,5], we find certain necessary and sufficient conditions on G such that Aut M N G be equal to Z Inn G ,…”

On $n^{th}$ class preserving automorphisms of $n$-isoclinism family

“…Zn(H) (H) such that Ψ(f ) = θ f , where θ f is the map defined in Lemma 3.1 (1). By Theorem 3.2, θ f ∈ Aut…”