# Intersection of subnormal subgroups

From Groupprops

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]

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 |

## Contents

## 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.