One-headed group

Symbol-free definition
A group is said to be one-headed if it has a proper normal subgroup that contains every proper defining ingredient::normal subgroup. Note that such a proper normal subgroup must also therefore be the unique defining ingredient::maximal normal subgroup. The quotient of the group by this maximal normal subgroup is termed the head of the group.

Note that simply saying that there is a unique maximal normal subgroup is a somewhat weaker statement, though it is equivalent for a group in which every proper subgroup is contained in a maximal subgroup.

Related properties

 * Monolithic group: This has a unique nontrivial normal subgroup contained in all the nontrivial normal subgroups.

Analogues in other algebraic structures

 * Local ring in the theory of commutative unital rings.
 * Local ring in the theory of noncommutative unital rings.