# Conjugate-join-closed subnormal implies join-transitively subnormal

From Groupprops

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., conjugate-join-closed subnormal subgroup) must also satisfy the second subgroup property (i.e., join-transitively subnormal subgroup)

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## Contents

## Statement

Suppose is a conjugate-join-closed subnormal subgroup of a group : for any subset of , the subgroup is a subnormal subgroup of .

Then, is a join-transitively subnormal subgroup of : for any subnormal subgroup of , the join of subgroups is also a subnormal subgroup of .

## Facts used

- Join of subnormal subgroups is subnormal iff their commutator is subnormal: Suppose are subnormal subgroups. Then is subnormal if and only if is subnormal, if and only if is subnormal.

## Related facts

- Join-transitively subnormal implies finite-automorph-join-closed subnormal
- Join-transitively subnormal of normal implies finite-conjugate-join-closed subnormal
- 2-subnormal implies join-transitively subnormal
- 2-subnormality is conjugate-join-closed

## Proof

**Given**: A conjugate-join-closed subnormal subgroup of a group , a subnormal subgroup of .

**To prove**: is subnormal.

**Proof**:

- (
**Given data used**: is conjugate-join-closed subnormal in ): Since is conjugate-join-closed subnormal in , is subnormal in . - (
**Given data used**: are subnormal in ;**Fact used**: fact (1)): By fact (1), we get that is subnormal in .