# Solvable not implies solvable automorphism group

From Groupprops

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

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## Statement

It is possible to have a solvable group such that the automorphism group of is not a solvable group.

## Proof

`Further information: endomorphism structure of elementary abelian group:E8`

Let be elementary abelian group:E8, which is a three-dimensional vector space over field:F2. The automorphism group is GL(3,2), which is a finite simple non-abelian group of order 168, hence is not solvable.