C-Characteristically Simple Groups

M. Shabani Attar

Bulletin of the Malaysian Mathematical Society, Tome 35 (2012) no. 1

Résumé

Let $G$ be a group and let $\mbox{Aut}_{c}(G)$ be the group of central automorphisms of $G$. We say that a subgroup $H$ of a group $G$ is c-characteristic if $\alpha(H)=H$ for all $\alpha \in \mbox{Aut}_{c}(G)$. 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 $Q_8$ where $Q_8$ is the quaternion group of order 8.

Détail

Classification : 20D45, 20D15.