Automorphism - Automorphism Group

If the automorphisms of an object X form a set (instead of a proper class), then they form a group under composition of morphisms. This group is called the automorphism group of X. That this is indeed a group is simple to see:

• Closure: composition of two endomorphisms is another endomorphism.
• Associativity: composition of morphisms is always associative.
• Identity: the identity is the identity morphism from an object to itself which exists by definition.
• Inverses: by definition every isomorphism has an inverse which is also an isomorphism, and since the inverse is also an endomorphism of the same object it is an automorphism.

The automorphism group of an object X in a category C is denoted Aut_C(X), or simply Aut(X) if the category is clear from context.