TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02

How statement and proof provenance work

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

Statement

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

Facts & Assumptions

Given: A group $G$ .

[L1]

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

[L2]

$Aut (G)$ 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]

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

[L5]

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

[L6]

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

Proof

technique · direct

1.1

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

L1L2L3L4L5L6givenalgebra

2.1

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

step 1.1L1L2L3L4L5L6givenalgebra

3.1

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

step 2.1∎

Depends on

Used by

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

The map g↦(x↦gxg−1)g↦(x↦gxg−1) is a homomorphism G→Aut⁡(G)G→Aut(G) with kernel Z(G)Z(G) and image Inn⁡(G)Inn(G)