# Holomorph of a ring

From Groupprops

## Definition

### Symbol-free definition

The holomorph of a ring is a group obtained as a semidirect product of its additive group by its multiplicative group of units, where the group of units acts by left multiplication (since the rings we consider for holomorph are usually commutative, we can omit the *left* qualifier).

Note that for a cyclic group, the holomorph of the ring is the same as the holomorph of the underlying additive group (because every automorphism of the additive group can be expressed as a multiplication).

### Definition with symbols

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## Group properties

- Metabelian group when the ring is commutative