# Minimal normal subgroup and core-free maximal subgroup need not be permutable complements

## Contents

## Statement

Suppose is a primitive group, is a core-free maximal subgroup of , and is a minimal normal subgroup of . Then, and need not be permutable complements. Specifically, either of these cases is possible:

- itself is a simple group and .
- is not a simple group, and is a proper subgroup of , but is trivial.

## Related facts

### Opposite facts

## Proof

### Example of situation where the whole group is simple

`Further information: alternating group:A5, A5 is simple`

Let be the alternating group on . Let . Let be the subgroup of comprising those permutations that fix .

Note that:

- is maximal, since it has index five.
- Since is simple and is a proper subgroup, is a core-free subgroup.
- Since is simple, is a minimal normal subgroup.

### Example of situation where the whole group is not simple

`Further information: symmetric group:S5`

Let be the symmetric group on . Let be the subgroup comprising those permutations that fix . Let be the subgroup comprising the even permutations, i.e., the alternating group on .

- is maximal, since it has index five, which is prime.
- is a core-free subgroup, since its conjugates are precisely the subgroups that fix the elements , and the intersection of all these is the identity element.
- is a minimal normal subgroup, since is a simple normal subgroup.