Intersection of subnormal subgroups

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
View other properties finitarily equivalent to subnormal subgroup | View other variations of subnormal subgroup |

Definition

A subgroup of a group is termed an intersection of subnormal subgroups if it satisfies the following equivalent conditions:

• It can be expressed as the intersection of a descending chain of subnormal subgroups.
• It can be expressed as an intersection of subnormal subgroups.

Note that since subnormality of fixed depth is closed under arbitrary intersections, these two definitions are equivalent.

Relation with other properties

Stronger properties

• Normal subgroup
• Subnormal subgroup

Weaker properties

• Descendant subgroup