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

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 G be a group and let N be a nonempty family of normal subgroups of G. Then

K:=⋂N∈NN

is a normal subgroup of G.

Facts & Assumptions

Given: A group G and a nonempty family N of normal subgroups of G.

[L1]

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

[L2]

A subgroup K≤G is normal if gKg−1⊆K for every g∈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∈NN is a subgroup of G.

L1
1.2

Fix g∈G and x∈K. For every N∈N, one has x∈N and N⊴G, so gxg−1∈N by [L2]. Hence gxg−1∈K.

givenL2
2.1

Thus gKg−1⊆K for every g∈G, and [L2] gives K⊴G.

step 1.1step 1.2L2∎

Depends on

Used by

Dependency tree · two levels

7 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