Characteristically simple group

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. Characteristically simple is a "weaker" condition than being a simple group, as simple groups must not have any proper nontrivial normal subgroups, which include characteristic subgroups.
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.
A minimal normal subgroup of a group "G" is a nontrivial normal subgroup "N" of "G" such that the only proper subgroup of "N" that is normal in "G" is the trivial subgroup. 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.