# Maximal implies normal or abnormal

This article gives the statement, and possibly proof, of a result according to which any Maximal subgroup (?) of a group satisfies exactly one of the following two subgroup properties: Normal subgroup (?) and Abnormal subgroup (?)

View other such statements

## Contents

## Statement

Any maximal subgroup of a group is either normal or abnormal.

## Related facts

### Weaker facts

## Definitions used

### Maximal subgroup

`Further information: Maximal subgroup`

A proper subgroup of a group is termed a **maximal subgroup** if there is no proper subgroup of properly containing . In other words, if is a subgroup of such that , then or .

### Normal subgroup

`Further information: Normal subgroup`

A subgroup of a group is termed **normal** in if it satisfies the following equivalent conditions:

- For any , the subgroup is equal to .
- The normalizer of in equals .

### Abnormal subgroup

`Further information: Abnormal subgroup`

A subgroup of a group is termed **abnormal** in if, for any , where .

## Proof

**Given**: A group , a maximal subgroup of .

**To prove**: is either normal or abnormal in .

**Proof**: Let be the normalizer of in . Then, . Thus, either , or . In the former case, is normal in .

In the latter case, . Now, pick any . Consider the subgroup . There are three cases:

- is not contained in : By maximality of , , so .
- : Thus, , and since , we get . Thus, .
- is a
*proper*subgroup of : is a proper subgroup of , forcing , which would imply that , a contradiction.