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
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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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- 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
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.