Characteristically simple group

Symbol-free definition
A nontrivial group is said to be characteristically simple if it satisfies the following equivalent conditions:


 * 1) It has no proper nontrivial defining ingredient::characteristic subgroup
 * 2) The defining ingredient::characteristic closure of any nontrivial subgroup is the whole group
 * 3) The defining ingredient::characteristic core of any proper subgroup is trivial

When the group is finite, this is equivalent to it being a direct product of pairwise isomorphic simple groups.

Definition with symbols
A nontrivial group $$G$$ is termed characteristically simple if it satisfies the following equivalent conditions:


 * 1) For any characteristic subgroup $$H$$ of $$G$$, $$H$$ is either trivial or the whole group
 * 2) For any nontrivial subgroup $$H$$ of $$G$$, the characteristic closure of $$H$$ (i.e., the subgroup generated by all $$\sigma(H)$$ for $$\sigma \in \operatorname{Aut}(G)$$), is the whole group $$G$$
 * 3) For any proper subgroup $$H$$ of $$G$$, the characteristic core of $$H$$ (i.e., the intersection of all $$\sigma(H)$$ for $$\sigma \in \operatorname{Aut}(G)$$), is the trivial subgroup (i.e., just the identity element)

When the group $$G$$ is finite, this is equivalent to $$G$$ being a direct product of pairwise isomorphic simple groups.

Formalisms
The group property of being characteristically simple is obtained by applying the simple group operator to the trim subgroup property of being characteristic.

Direct products
The direct product of two characteristically simple groups is characteristically simple if and only if they are powers of the same simple group. Note that the simple group is unique upto isomorphism.

Textbook references

 * , Page 25
 * , Page 16 (definition in paragraph, preceding Theorem 1.4)