A group automorphism is an isomorphism from a group to itself. If is a finite multiplicative group, an automorphism of can be described as a way of rewriting its multiplication table without altering its pattern of repeated elements. For example, the multiplication table of the group of 4th roots of unity can be written as shown above, which means that the map defined by

is an automorphism of .

The map is also a group automorphism for as long as is not congruent to 0. Conjugating by a fixed element is a group automorphism called an inner automorphism.

In general, the automorphism group of an algebraic object , like a ring or field, is the set of isomorphisms of that object , and is denoted . It forms a group by composition of maps. For a fixed group , the collection of group automorphisms is the automorphism group .