




Volume 35 • Number 1 • 2012 

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CCharacteristically Simple Groups
M. Shabani Attar
Abstract.
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 ccharacteristic if $\alpha(H)=H$ for all $\alpha \in \mbox{Aut}_{c}(G)$. We say that a group $G$ is ccharacteristically simple group if it has no nontrivial ccharacteristic subgroup. If every subgroup of $G$ is ccharacteristic then $G$ is called coDedekindian group. In this paper we characterize ccharacteristically simple groups. Also if $G$ is a direct product of two groups $A$ and $B$ we study the relationship between the coDedekindianness of $G$ and the coDedekindianness of $A$ and $B$. We prove that if $G$ is a coDedekindian finite nonabelian 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.
2010 Mathematics Subject Classification: 20D45, 20D15.
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