# 2-subnormality is strongly intersection-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., 2-subnormal subgroup) satisfying a subgroup metaproperty (i.e., strongly intersection-closed subgroup property)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about 2-subnormal subgroup |Get facts that use property satisfaction of 2-subnormal subgroup | Get facts that use property satisfaction of 2-subnormal subgroup|Get more facts about strongly intersection-closed subgroup property

## Contents

## Statement

An arbitrary intersection of 2-subnormal subgroups of a group is 2-subnormal.

## Related facts

- Normality is strongly intersection-closed
- Subnormality of bounded depth is strongly intersection-closed
- Subnormality is not intersection-closed
- Normality is strongly UL-intersection-closed

## Facts used

- Normality is strongly intersection-closed: An arbitrary intersection of normal subgroups is normal.
- Normality is strongly UL-intersection-closed: if are subgroups for and is normal in for each , then the intersection of the s is normal in the intersection of the s.

## Proof

**Given**: A group , a collection of 2-subnormal subgroups of .

**To prove**: The intersection is also a 2-subnormal subgroup of .

**Proof**: Let be the normal closure of in . Thus, each is normal in . Note that, by the definition of 2-subnormality, is normal in for each .

Let be the intersection of the s and be the intersection of the . By fact (1), is normal in . By fact (2), is normal in . Thus, is 2-subnormal in .

(Note that is not necessarily the normal closure of in -- we can only say that it contains the normal closure of in .)