Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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.

Correspondence theorem: subgroups of G/NG/N correspond to subgroups of GG containing NN, with normality preserved

Statement

Correspondence theorem: subgroups of G/NG/N correspond to subgroups of GG containing NN, with normality preserved.

For N⊴GN\mathrel{\trianglelefteq}G, the maps H↦H/NH\mapsto H/N and K↦π−1(K)K\mapsto\pi^{-1}(K) are inverse inclusion-preserving bijections between subgroups HH with N≤H≤GN\le H\le G and subgroups K≤G/NK\le G/N; they preserve normality.

Facts & Assumptions

Given: A normal subgroup N⊴GN\mathrel{\trianglelefteq}G and the quotient map π:G→G/N\pi:G\to G/N.

[L1]

π\pi is surjective with kernel NN (The canonical projection π:G→G/N\pi:G\to G/N, π(g)=gN\pi(g)=gN, is a surjective group homomorphism).

[L2]

Kernels and images are defined by inverse images and values (The kernel and image of a group homomorphism).

[L3]

Images are subgroups and kernels are normal (The image of a group homomorphism is a subgroup and its kernel is a normal subgroup).

[L4]

The subgroup criterion is closure under xy−1xy^{-1} (One-step subgroup test: a nonempty H⊆GH \subseteq G is a subgroup iff gh−1∈Hgh^{-1} \in H for all g,h∈Hg, h \in H; the identity and the inverses of HH are then those of GG).

[L5]

Normality has the conjugation and coset characterisations (Equivalent characterisations of a normal subgroup by conjugates and left and right cosets).

[L6]

G/NG/N is the quotient group of cosets (The quotient group G/NG/N and coset product (gN)(hN)=ghN(gN)(hN)=ghN).

Proof

technique · direct
1.1

For N≤H≤GN\le H\le G, H/N=π[H]H/N=\pi[H] is a subgroup, while π−1(K)\pi^{-1}(K) is a subgroup containing ker⁡π=N\ker\pi=N.

L1L2L3L4L5L6givenconstruct
2.1

Surjectivity gives π[π−1(K)]=K\pi[\pi^{-1}(K)]=K, and N≤HN\le H gives π−1(π[H])=H\pi^{-1}(\pi[H])=H; both assignments therefore preserve inclusion and are inverse.

step 1.1L1L2L3L4L5L6givenalgebra
3.1

The image and preimage calculation of step 2.1 also preserves normality.

step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 37 results over 15 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