TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The map $g\mapsto(x\mapsto gxg^{-1})$ is a homomorphism $G\to\operatorname{Aut}(G)$ with kernel $Z(G)$ and image $\operatorname{Inn}(G)$

Statement

The map $g\mapsto(x\mapsto gxg^{-1})$ is a homomorphism $G\to\operatorname{Aut}(G)$ with kernel $Z(G)$ and image $\operatorname{Inn}(G)$ .

Facts & Assumptions

Given: A group $G$ .

[L1]

$\operatorname{Inn}(G)=\{c_g:g\in G\}$ (Inner automorphisms and $\operatorname{Inn}(G)$ ).

[L2]

$\operatorname{Aut}(G)$ is a group under composition (The automorphisms of a group form a group under composition).

[L3]

Kernels and images are defined for group homomorphisms (The kernel and image of a group homomorphism).

[L4]

$Z(G)=\{g\in G:gx=xg\text{ for every }x\in G\}$ (The center $Z(G)$ of a group).

[L5]

A homomorphism preserves the group operation (Monoid homomorphism and group homomorphism).

[L6]

$(gh)^{-1}=h^{-1}g^{-1}$ (In a group $e^{-1} = e$ , $(g^{-1})^{-1} = g$ and $(gh)^{-1} = h^{-1}g^{-1}$ , the order of the last product being essential).

Proof

technique · direct

1.1

Define $C(g)=c_g$ ; for every $x$ , $C(gh)(x)=ghx(gh)^{-1}=g(hxh^{-1})g^{-1}=(C(g)\circ C(h))(x)$ .

L1L2L3L4L5L6givenalgebra

2.1

Now $C(g)=\operatorname{id}_G$ exactly when $gxg^{-1}=x$ for every $x$ , equivalently $g\in Z(G)$ , and its image is $\operatorname{Inn}(G)$ by definition.

step 1.1L1L2L3L4L5L6givenalgebra

3.1

Thus $C$ is a homomorphism with kernel $Z(G)$ and image $\operatorname{Inn}(G)$ .

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Milne, Group Theory, Automorphisms of Groups (standard reference, not scraped)

The map g↦(x↦gxg−1)g\mapsto(x\mapsto gxg^{-1})g↦(x↦gxg−1) is a homomorphism G→Aut⁡(G)G\to\operatorname{Aut}(G)G→Aut(G) with kernel Z(G)Z(G)Z(G) and image Inn⁡(G)\operatorname{Inn}(G)Inn(G)