# Complement to normal subgroup is isomorphic to quotient group

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## Contents

## Statement

Suppose is a group, is a normal subgroup and is a subgroup such that and are permutable complements: and is trivial. Then, .

In particular, any two permutable complements to are isomorphic to each other.

## Related facts

### Caveats

Note the following:

- If is a normal subgroup of , it is not necessary that has a permutable complement in . For instance, if is the cyclic group of order four and is a subgroup of order two, does not have a permutable complement. A normal subgroup that
*does*have a permutable complement is termed a complemented normal subgroup. A subgroup that occurs as the permutable complement to a normal subgroup is termed a retract. - Any lattice complement to in is also a permutable complement to in . This is because any normal subgroup is permutable: its product with any subgroup is a subgroup.
`Further information: Equivalence of definitions of complemented normal subgroup`

- Complements to abelian normal subgroup are automorphic: If is an abelian normal subgroup of and are two permutable complements to in , then there is an automorphism of that is the identity map on and sends to . In particular, and are automorphic subgroups.
- Complements to normal subgroup need not be automorphic: If is a normal subgroup of and are two permutable complements to in , there need not be an automorphism of sending to .
- Retract not implies normal complements are isomorphic: This states that if and are two permutable complements to a subgroup , and both and are normal, this does
*not*imply that is isomorphic to . In other words, interchanging the role of which subgroup is the normal one renders the result false. - Schur-Zassenhaus theorem (normal Hall implies permutably complemented and Hall retract implies order-conjugate): This states that every normal Hall subgroup, and hence every normal Sylow subgroup, has a complement, and that any two such complements are conjugate subgroups in the whole group.
- Every group of given order is a permutable complement for symmetric groups: Any group of order occurs as a permutable complement to in , via the embedding obtained by Cayley's theorem. This is a far cry from the fact that any two permutable complements to a normal subgroup are isomorphic.
- Semidirect product is not left-cancellative for finite groups: We can have finite groups such that , but is not isomorphic to . Note that the isomorphism between and must
*not*map the normal subgroup to the normal subgroup , otherwise we would contradict the result of this page.

## Proof

### Proof using second isomorphism theorem

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