# 2-subnormal implies conjugate-join-closed subnormal

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., 2-subnormal subgroup) must also satisfy the second subgroup property (i.e., conjugate-join-closed subnormal subgroup)
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## Statement

A 2-subnormal subgroup of a group is a conjugate-join-closed subnormal subgroup: the join of an arbitrary number of conjugate subgroups to a 2-subnormal subgroup is subnormal.

## Facts used

1. 2-subnormality is conjugate-join-closed

## Proof

The proof follows directly from fact (1), which states that an arbitrary join of conjugate 2-subnormal subgroups is 2-subnormal. Since 2-subnormal subgroups are subnormal, the conclusion follows.