of a group
An automorphism such that
for a certain fixed element . The set of all inner automorphisms of forms a normal subgroup in the group of all automorphisms of ; this subgroup is isomorphic to , where is the centre of (cf. Centre of a group). Automorphisms that are not inner are called outer automorphisms.
Other relevant concepts include those of an inner automorphism of a monoid (a semi-group with a unit element) and an inner automorphism of a ring (associative with a unit element), which are introduced in a similar way using invertible elements.
Let be a Lie algebra and an element of for which is nilpotent. Then
defines an automorphism of . Such an automorphism is called an inner automorphism of . More generally, the elements in the group generated by them are called inner automorphisms. It is a normal subgroup of .
If is a real or complex Lie group with semi-simple Lie algebra, then the inner automorphisms constitute precisely the identity component of the group of automorphisms of .
|[a1]||M. Hall jr., "The theory of groups" , Macmillan (1959)|
|[a2]||J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972) pp. §5.4|
|[a3]||J.-P. Serre, "Algèbres de Lie semi-simples complexes" , Benjamin (1966)|
Inner automorphism. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Inner_automorphism&oldid=15849