Simple subnormal subgroup

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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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View a complete list of semi-basic definitions on this wiki

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:

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

Weaker properties

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.