Some properties on $$\mathrm{IA_Z}$$-automorphisms of groups

Let G be a group and $$\mathrm{IA}(G)$$ IA ( G ) denote the group of all automorphisms of G , which induce identity map on the abelianized group $$G_{ab}=G/G'$$ G ab = G / G ′ . Also the group of those $$\mathrm{IA}$$ IA -automorphisms which fix the centre element-wise is denoted by $$\mathrm{IA_Z}(G)$$ IA Z ( G ) . In the present article, among other results and under some condition we prove that the derived subgroups of finite p -groups, for which $$\mathrm{IA_Z}$$ IA Z -automorphisms are the same as central automorphisms, are either cyclic or elementary abelian.