# Simple subnormal subgroup

From Groupprops

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 definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated to Simple subnormal subgroup, all facts related to Simple subnormal subgroup) |Survey articles about this | Survey articles about definitions built on this

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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

### 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 transitiveABOUT 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.