# Conjugate-permutability satisfies intermediate subgroup condition

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This article gives the statement, and possibly proof, of a subgroup property (i.e., conjugate-permutable subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)

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## Statement

### Verbal statement

Any conjugate-permutable subgroup of a group is also conjugate-permutable in every intermediate subgroup.

### Statement with symbols

If is a conjugate-permutable subgroup of a group , and , then is also conjugate-permutable in .

## Related facts

- Permutability satisfies intermediate subgroup condition
- Permutability satisfies transfer condition
- Permutability satisfies inverse image condition
- Normality satisfies intermediate subgroup condition
- Normality satisfies transfer condition
- Normality satisfies inverse image condition
- Automorph-permutability does not satisfy intermediate subgroup condition

### Applications

## Proof

**Given**: A group , a subgroup with the property that for all . .

**To prove**: for all .

**Proof**: Since and , we have , and the statement to prove follows directly from the given data.