CorollaryStatement: 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.

$G / Z (G) ≅ Inn (G)$

Statement

$G / Z (G) ≅ Inn (G)$ .

Facts & Assumptions

Given: A group $G$ .

[L1]

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

[L2]

$G / \ker f ≅ im f$ for every group homomorphism $f$ (First isomorphism theorem for groups: $G / \ker f ≅ im f$ ).

[L3]

$Z (G)$ is the center of $G$ (The center $Z (G)$ of a group).

Proof

technique · direct

1.1

Apply [L2] to the conjugation homomorphism of [L1].

L1L2L3given

2.1

Substituting its kernel and image yields $G / Z (G) ≅ Inn (G)$ .

step 1.1L1L2L3given

3.1

This is the asserted isomorphism.

step 2.1∎

Depends on

Used by

An automorphism of an extraspecial p-group acting trivially on its Frattini quotient is inner Proposition

Dependency tree · two levels

14 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)

G/Z(G)≅Inn⁡(G)G/Z(G)≅Inn(G)