# Abelian minimal normal subgroup and core-free maximal subgroup are permutable complements

## Contents

## Statement

Suppose is a primitive group with core-free maximal subgroup . Then, if has an Abelian minimal normal subgroup , the following are true:

- and intersect trivially
- The induced action of on by left multiplication, is equivalent to the regular action of a group on itself.

## Definitions used

### Core-free subgroup

`Further information: Core-free subgroup`

A subgroup of a group is termed core-free if its normal core in the whole group is trivial, or equivalently, if it does not contain any nontrivial normal subgroup of the whole group.

### Maximal subgroup

`Further information: Maximal subgroup`

A maximal subgroup of a group is a proper subgroup that is not contained in any bigger proper subgroup.

### Primitive group

`Further information: Primitive group`
A primitive group is a group that possesses a core-free maximal subgroup.

## Facts used

- Abelian normal subgroup and core-free subgroup generate whole group implies they intersect trivially

## Related facts

### Plinth theorem

`Further information: plinth theorem`

The plinth theorem is a similar result to the case of two distinct minimal nomral subgroups.

### More results under similar hypotheses

Proceeding further along the lines of this proof, one can show that:

- Abelian permutable complement to core-free subgroup is self-centralizing
- Primitive implies Fitting-free or elementary Abelian Fitting subgroup: In other words, is maximal among Abelian normal subgroups, and
*also*is the*only*nontrivial Abelian normal subgroup, and is the*only*nontrivial nilpotent normal subgroup.

### Breakdown of similar results

## Proof

**Given**: A primitive group with core-free maximal subgroup , and an Abelian minimal normal subgroup

**To prove**: and are permutable complements: and is trivial

**Proof**: Since is normal, is a subgroup of containing . Since is maximal, either or . If , then , but this cannot happen because does not contain any nontrivial normal subgroup of . Hence .

Thus is an Abelian normal subgroup and is a core-free subgroup that together generate . Applying the fact (1) stated above, is trivial, so and are permutable complements.