# Maximal conjugate-permutable implies normal

From Groupprops

This article gives the proof of a maximality equivalence. In other words, there are two subgroup properties: a stronger one (Normal subgroup (?)) and a weaker one (Conjugate-permutable subgroup (?)). However, any subgroupmaximalamong the proper subgroups with the weaker property also has the stronger property, and is thus also maximal among proper subgroups with the stronger property.

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## Statement

Suppose is a proper Conjugate-permutable subgroup (?) of a group such that is not properly contained in any proper conjugate-permutable subgroup of . Then, is a Normal subgroup (?) of .

## Related facts

### Similar facts

- Pronormal implies self-conjugate-permutable
- Maximal implies self-conjugate-permutable
- Conjugate-permutable and self-conjugate-permutable implies normal

### Applications

## Facts used

- Conjugate-permutability is conjugate-join-closed: If is a conjugate-permutable subgroup of , and , then is also a conjugate-permutable subgroup of .
- Product of conjugates is proper: If is a proper subgroup of , and , then is a proper subset of .

## Proof

**Given**: A group , a proper conjugate-permutable subgroup of such that is not contained in any proper conjugate-permutable subgroup of .

**To prove**: is normal in : for any , .

**Proof**:

- (
**Fact used**: fact (1), conjugate-permutability is conjugate-join-closed): Suppose . Then, since is conjugate-permutable in , fact (1) tells us that is also conjugate-permutable in . - (
**Given data used**: is*maximal*conjugate-permutable): By our assumption, either or . - (
**Fact used**: fact (2), product of conjugates is proper): If , we have , which yields a contradiction by fact (2). - Combining steps (2) and (3), we see that , forcing , as required.