Group theory and generalizations Definition

Inner automorphism

Builds on Automorphism group Conjugate elements

Definition

Let $G$ be a group and let $g \in G$ . The inner automorphism given by $g$ is the map $c_g \colon G \to G$ , $c_g(x) = gxg^{-1}$ . The set of all inner automorphisms of $G$ is written $\operatorname{Inn}(G)$ .

Remarks

Each $c_g$ is an automorphism of $G$ , with inverse $c_{g^{-1}}$ . Since $c_g \circ c_h = c_{gh}$ , the map $g \mapsto c_g$ is a homomorphism from $G$ to $\operatorname{Aut}(G)$ , and $\operatorname{Inn}(G)$ is its image. The subgroup $\operatorname{Inn}(G)$ is normal in $\operatorname{Aut}(G)$ , because $\alpha \circ c_g \circ \alpha^{-1} = c_{\alpha(g)}$ for each automorphism $\alpha$ . In an abelian group, each inner automorphism is the identity map.

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Group concept
Conjugate elements definition
Group homomorphism definition
Group isomorphism definition
Automorphism group definition
Inner automorphism definition

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3 pages build on this one.

Tag	`0H47DMFN3S`
URI	`m2://0H47DMFN3S`
Aliases	`conjugation-automorphism`
Kind	definition
Writing	W1 stated
MSC	20F28
Defines	Inner automorphism
Uses	Automorphism group, Conjugate elements

Inner automorphism

Definition

Remarks

Reading path

Where this leads

Further reading

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