Simple subnormal subgroup

Symbol-free definition
A subgroup of a group is termed a simple subnormal subgroup or a minimal subnormal subgroup if it satisfies the following equivalent conditions:


 * It is simple as an abstract group and subnormal as a subgroup
 * It is minimal among subnormal subgroups, viz there is no smaller nontrivial subgroup of it that is subnormal in the whole group

This is essentially the second equivalent formulation.

Stronger properties

 * Simple normal subgroup

Weaker properties

 * Component
 * Subnormal subgroup

Metaproperties
In fact, the only simple subnormal subgroup of a simple subnormal subgroup is itself.