Characteristically Simple

Definition

A group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups.

Proposition

Any characteristically simple group is a product of isomorphic simple groups.

\begin{proof} Let $T$ be a minimal normal subgroup of $G$. Then $T$ is isomorphic to a direct product of isomorphic simple groups, see here. Define $M=⟨T^{\varphi }:\varphi \in \mathrm{Aut}(G)⟩$. Note that $M$ is a characteristic group of $G$ and so $M=G$. Since $T^{\varphi }$ is normal in $G$ and $T^{\varphi }\cap T\in \left\{1,T\right\}$ by $T$ minimal normal subgroup, we have $M\cong T\times \cdots \times T$ by this lemma. Therefore, $G$ is a product of isomorphic simple groups. \end{proof}