Groups with some central automorphisms fixing the central kernel quotient

Let G be a group. An automorphism \alpha of a group G is called a central automorphism, if x^{-۱}x^{\alpha}\in Z(G) for all x\in G. Let L_c(G) be the central kernel of G, that is the set of elements of G fixed by all central  automorphisms of G and Aut_{L_c}(G) denote the group of all central automorphisms of G fixing G/L_c(G) element-wise. In the present paper, we investigate the properties of such automorphisms. Moreover, a full classification of p-groups G of order at most p^۵ where Aut_{L_c}(G)=Inn(G) is also given.

Keywords:

automorphism group , central kernel , central autocommutator