Characteristically simple group

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In mathematics, in the field of group theory, a group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups. Characteristically simple groups are sometimes also termed elementary groups.

A finite group is characteristically simple if and only if it is the direct product of isomorphic simple groups. In particular, a finite solvable group is characteristically simple if and only if it is an elementary abelian group. This does not hold in general for infinite groups; for example, the rational numbers form a characteristically simple group that is not a direct product of simple groups.

Every minimal normal subgroup of a group is characteristically simple. This follows from the fact that a characteristic subgroup of a normal subgroup is normal.