Editing Automorphism (section)

==Inner and outer automorphisms==
{{main article|Inner automorphism|Outer automorphism group}}
In some categories—notably [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[Lie algebra]]s—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.

In the case of groups, the [[inner automorphism]]s are the conjugations by the elements of the group itself. For each element ''a'' of a group ''G'', conjugation by ''a'' is the operation {{nowrap|''φ''<sub>''a''</sub> : ''G'' → ''G''}} given by {{nowrap|1=''φ''<sub>''a''</sub>(''g'') = ''aga''<sup>−1</sup>}} (or ''a''<sup>−1</sup>''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 automorphism]]s. The [[quotient group]] {{nowrap|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 algebra|unital]] [[ring (mathematics)|ring]] or [[algebra over a field|algebra]] where ''a'' is any [[Unit (ring theory)|invertible element]]. For [[Lie algebra]]s the definition is slightly different.