Alphabeta Math
LemmaStatement: AI-adaptedProof: AI-generatedSession-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.

The intersection of a nonempty family of normal subgroups is normal

Statement

Let GG be a group and let N\mathcal N be a nonempty family of normal subgroups of GG. Then

K:=⋂N∈NNK:=\bigcap_{N\in\mathcal N}N

is a normal subgroup of GG.

Facts & Assumptions

Given: A group GG and a nonempty family N\mathcal N of normal subgroups of GG.

[L1]

The intersection of a nonempty family of subgroups of a group is a subgroup (The intersection of a nonempty family of subgroups of GG is a subgroup of GG).

[L2]

A subgroup K≤GK\le G is normal if gKg−1⊆KgKg^{-1}\subseteq K for every g∈Gg\in G (Equivalent characterisations of a normal subgroup by conjugates and left and right cosets).

Proof

technique · direct
1.1

By [L1], the set K=⋂N∈NNK=\bigcap_{N\in\mathcal N}N is a subgroup of GG.

L1
1.2

Fix g∈Gg\in G and x∈Kx\in K. For every N∈NN\in\mathcal N, one has x∈Nx\in N and N⊴GN\mathrel{\trianglelefteq}G, so gxg−1∈Ngxg^{-1}\in N by [L2]. Hence gxg−1∈Kgxg^{-1}\in K.

givenL2
2.1

Thus gKg−1⊆KgKg^{-1}\subseteq K for every g∈Gg\in G, and [L2] gives K⊴GK\mathrel{\trianglelefteq}G.

step 1.1step 1.2L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 10 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