**Inner and Outer Automorphisms**

In some categories—notably groups, rings, and Lie algebras—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.

In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself. For each element *a* of a group *G*, conjugation by *a* is the operation φ_{a} : *G* → *G* given by (or *a*−1*ga*; usage varies). 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 other automorphisms are called outer automorphisms. 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. For Lie algebras the definition is slightly different.

Read more about this topic: Automorphism

### Famous quotes containing the word outer:

“The guarantee that our self enjoys an intended relation to the *outer* world is most, if not all, we ask from religion. God is the self projected onto reality by our natural and necessary optimism. He is the not-me personified.”

—John Updike (b. 1932)