On the Inner Automorphisms and Central Automorphisms of Nilpotent Group of Class 2 Which Fix the Centre Elementwise

Abstract

Suppose that G is a finite p-group. It was shown (see [2]) that C^∗ the set of all central automorphisms of G which elementarily fixes the centre of G elementwise, is isomorphic to the group of all Inner automorphisms of G if and only if G is abelian or G is nilpotent of class 2 for which the centre of G is cyclic. More so, if G is finitely generated then G can be represented in a particular simple form (see [3]). Moreover, suppose that G is a finite p-group such that Aut(G) ≡ E_pm . Then , C_Autc (G)(Z(G)) = G/Z(G) .