Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04
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FALSE: Schur-Zassenhaus says every Hall subgroup is normal

Statement

Schur-Zassenhaus says that every Hall subgroup of a finite group is normal.

Facts & Assumptions

Given: The subgroup ⟨(12)⟩≤S3.

[L1]

A Hall subgroup is defined by a coprime order-index condition (Hall pi-subgroup).

[L2]

Schur-Zassenhaus starts from a normal Hall subgroup and then produces a complement (Schur-Zassenhaus existence theorem).

Refutation

technique · direct
1.1

The subgroup ⟨(12)⟩ has order 2 and index 3, so [L1] makes it a Hall {2}-subgroup of S3.

givenL1
2.1

It is not normal, because (123)(12)(123)−1=(23)∉⟨(12)⟩. Thus Hall subgroups need not be normal. The actual theorem [L2] assumes normality of the Hall subgroup as a hypothesis, so the claim is false.

L2step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources