# Holomorph of elementary abelian group not implies every automorphism is inner

From Groupprops

## Contents

## Statement

It is possible to have an Elementary abelian group (?) such that the holomorph of is not a Complete group (?). In fact, we can construct examples where the holomorph is *not* a Group in which every automorphism is inner (?).

The smallest example is the elementary abelian group of order eight.

## Related facts

### Opposite facts for abelian case

- Holomorph of cyclic group of odd prime order is complete: This in turn can be deduced from the fact that semidirect product with self-normalizing subgroup of automorphism group of coprime order implies every automorphism is inner.

### Analogue for non-abelian case

- Additive group of a field implies monolith in holomorph, additive group of a field implies characteristic in holomorph
- Odd-order elementary abelian group is fully invariant in holomorph
- Odd-order cyclic group equals commutator subgroup of holomorph, odd-order cyclic group is fully invariant in holomorph, odd-order cyclic group is characteristic in holomorph
- Odd-order abelian group not is fully invariant in holomorph