# Product of conjugates is proper

This article describes an easy-to-prove fact about basic notions in group theory, that is not very well-known or important in itself

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*This article describes a result of the form that argues that a subset constructed in a certain fashion is proper, viz it is not the whole group*

## Contents

## Statement

### Verbal statement

Given any two proper subgroups of a group that are conjugate to each other, their product is a proper subset of the group.

### Symbolic statement

Let be a proper subgroup, and let be a conjugate of . Then is a proper subset of .

## Related facts

- Union of all conjugates of subgroup of finite index is proper: This states that the union of all conjugates of a proper subgroup in a finite group is again proper.

### Applications and similar facts

- Maximal implies normal or abnormal: The proof idea here is very similar.
- Maximal conjugate-permutable implies normal: This is an easy corollary.
- Maximal implies self-conjugate-permutable
- Conjugate-permutable and self-conjugate-permutable implies normal
- Conjugate-permutable implies subnormal in finite

## Proof

**Given**: A finite group , two proper conjugate subgroups and , where .

**To prove**: is a proper subset of .

**Proof**: Suppose not, i.e., suppose .

- , so we can write where .
- Thus, . This yields .
- But we know that , so we get .
- We thus get , contradicting the assumption that is a proper subgroup of .