Simple subnormal subgroup

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): simple group
View a complete list of such conjunctions

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition

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

In terms of the minimal operator

This property is obtained by applying the minimal operator to the property: nontrivial subnormal subgroup
View other properties obtained by applying the minimal operator

This is essentially the second equivalent formulation.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of simple subnormal subgroup

Relation with other properties

Stronger properties

• Simple normal subgroup

Weaker properties

• Component
• Subnormal subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

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