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

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G/Z(G)≅Inn⁡(G)G/Z(G)\cong\operatorname{Inn}(G)

Statement

G/Z(G)≅Inn⁡(G)G/Z(G)\cong\operatorname{Inn}(G).

Facts & Assumptions

Given: A group GG.

[L1]

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

[L2]

G/ker⁡f≅im⁡fG/\ker f\cong\operatorname{im}f for every group homomorphism ff (First isomorphism theorem for groups: G/ker⁡f≅im⁡fG/\ker f\cong\operatorname{im}f).

[L3]

Z(G)Z(G) is the center of GG (The center Z(G)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)G/Z(G)\cong\operatorname{Inn}(G).

step 1.1L1L2L3given
3.1

This is the asserted isomorphism.

step 2.1∎

Depends on

Used by

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