# Outer automorphism group

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## Definition

Let be a group. The **outer automorphism group** of , denoted , is defined as the quotient group of the automorphism group by the inner automorphism group .

Elements of the outer automorphism group are sometimes termed *outer automorphism classes*. Each outer automorphism class is a coset of the inner automorphism group -- it is *not* a single automorphism. Thus, the outer automorphism group is *not* the set of outer automorphisms.