# Complements to normal subgroup need not be automorphic

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Revision as of 18:39, 8 November 2008 by Vipul (talk | contribs) (New page: ==Statement== Suppose <math>G</math> is a group, <math>N</math> is a normal subgroup, and <math>H</math> and <math>K</math> are permutable complements to <math>N</math> in <math>G</ma...)

## Statement

Suppose is a group, is a normal subgroup, and and are permutable complements to in . Then, it is *not* necessary that there exists an automorphism of sending to .

## Related facts

- Schur-Zassenhaus theorem
- Complements to Abelian normal subgroup are automorphic
- Complement to normal subgroup is isomorphic to quotient

## Proof

### A generic example

Let be any non-Abelian group. Consider and the subgroup . Let be the subgroup and be the subgroup .

Note that:

- is normal in : In fact, it is a direct factor of .
- is a permutable complement to in .
- is a permutable complement to in .
- is normal in : In fact, it is a direct factor of .
- is not normal in : Pick such that do not commute. Then, we have . Thus, a conjugate of an element in lies outside .