C-Characteristically Simple Groups

Let G be a group and let Autc(G) be the group of central automorphisms of G. We say that a subgroup H of a group G is c-characteristic if α(H)=H for all αϵAutc(G). We say that a group G is c-characteristically simple group if it has no non-trivial c-characteristic subgroup. If every subgroup of G is c-characteristic then G is called co-Dedekindian group. In this paper we characterize c-characteristically simple groups. Also if G is a direct product of two groups A and B we study the relationship between the co-Dedekindianness of G and the co-Dedekindianness of A and B. We prove that if G is a co-Dedekindian finite non-abelian group, then G is Dedekindian if and only if G is isomorphic to Q8 where Q8 is the quaternion group of order 8.