# Inner automorphism

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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.