# Solvable automorphism group implies solvable of derived length at most one more

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group whose automorphism group is solvable) must also satisfy the second group property (i.e., solvable group)

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This article gives a result about how information about the structure of the automorphism group of a group (abstractly, or in action) can control the structure of the group

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## Statement

Suppose is an group whose automorphism group is solvable, i.e., a group whose automorphism group is solvable. Then, is also solvable. Moreover, if the derived length of the automorphism group of is , the derived length of is at most .

## Related facts

### Similar facts

- Aut-abelian implies class two
- Aut-nilpotent implies nilpotent of class at most one more
- Trivial automorphism group implies trivial or order two

### Converse

The converse is not true: solvable not implies aut-solvable.

## Facts used

- Group acts as automorphisms by conjugation
- Solvability is subgroup-closed
- The center of a group is abelian, hence solvable
- Solvability is extension-closed