# Cofactorial automorphism

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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## Definition

### For a finite group

An automorphism $\sigma$ of a finite group $G$ is termed a cofactorial automorphism for $G$ if every prime dividing the order of $\sigma$ (viewed as an element of the automorphism group of $G$) also divides the order of $G$.

### For a periodic group

An automorphism $\sigma$ of a periodic group $G$ is termed a cofactorial automorphism for $G$ if every prime dividing the order of $\sigma$ equals the order of some non-identity element of $G$.

### For a group with elements of infinite order

If the group has any element of infinite order, then every automorphism of the group is considered to be a cofactorial automorphism.