Inner automorphism
of a group
An automorphism such that
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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.
Comments
Let be a Lie algebra and
an element of
for which
is nilpotent. Then
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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
.
References
[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://encyclopediaofmath.org/index.php?title=Inner_automorphism&oldid=35114