# Complements to abelian normal subgroup are automorphic

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## Contents

## Statement

Suppose is a group, is an Abelian normal subgroup, and are permutable complements to in . Then, there is an automorphism of such that:

- The restriction of to is the identity map on .
- induces an isomorphism from to .

In particular, and are automorphic subgroups.

(Note: There may exist automorphic subgroups to that are *not* permutable complements to ).

## Related facts

- Complement to normal subgroup is isomorphic to quotient
- Hall retract implies order-conjugate: This is the conjugacy part of the Schur-Zassenhaus theorem, and states that, for a normal Hall subgroup, any two complements are not just automorphic, they are also conjugate subgroups.
- Complements to normal subgroup need not be automorphic

### Breakdown when the normality constraint is removed or shifted

- Every group of given order is a permutable complement for symmetric groups: A complete breakdown of the analogous statement when the subgroup is no longer assumed to be normal.
- Retract not implies every permutable complement is normal
- Retract not implies normal complements are isomorphic