Automorphism

It is, loosely speaking, the symmetry group of the object.

In an algebraic structure such as a group, a ring, or vector space, an automorphism is simply a bijective homomorphism of an object into itself.

For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply the identity function, and is often called the trivial automorphism.

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

One can easily check that conjugation by a is a group automorphism.

The inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma.

The quotient group Aut(G) / Inn(G) is usually denoted by Out(G); the non-trivial elements are the cosets that contain the outer automorphisms.

The same definition holds in any unital ring or algebra where a is any invertible element.