Conjugation by an element

Inner group automorphism

The inner group automorphism is a normal subgroup given by conjugation by an element, group i.e. if then for some . It is hence the image of conjugation as a group action .

Proof of normal subgroup

That is a subgroup follows from the fact that it is the image of the homomorphism . Let and . Then for any

thus .

By the First isomorphism theorem, this is isomorphic to , where the divisor is the centre.

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