Alphabeta Math
LemmaStatement: Literature-sourcedProof: Literature-sourcedprecheck passaudited 2026-08-13
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.

Characteristic subgroups are normal, and characteristicity is transitive

Statement

If Hchar⁡G, then H⊴G. If Kchar⁡H and Hchar⁡G, then Kchar⁡G.

Facts & Assumptions

Given: Groups and subgroups satisfying the hypotheses of either assertion.

[F1]

Hchar⁡G means that every automorphism of G maps H onto itself (Characteristic subgroups).

[F2]

H⊴G means that conjugation by every element of G preserves H (Normal subgroup: invariance under conjugation).

Proof

technique · direct
1.1

For each g∈G, conjugation x↦gxg−1 is an automorphism of G; if Hchar⁡G, [F1] says it preserves H, so [F2] gives H⊴G.

F1F2
1.2

Suppose Kchar⁡Hchar⁡G and let α∈Aut⁡(G). By [F1], α(H)=H, so α∣H is an automorphism of H; applying [F1] to Kchar⁡H gives α(K)=K.

F1
2.1

Since step 1.2 holds for every automorphism of G, Kchar⁡G; together with step 1.1 this proves both assertions.

step 1.1step 1.2F1∎

Depends on

Used by

Dependency tree · two levels

5 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