{{Short description|Group without proper nontrivial characteristic subgroups}} 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 a 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.
== References ==
*{{Citation | last1=Robinson | first1=Derek John Scott | title=A course in the theory of groups | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-94461-6 | year=1996}}
Category:Properties of groups
{{group-theory-stub}}