False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-17

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False statement: one unique Sylow subgroup forces the whole group to be a direct product

Statement

False claim: one unique Sylow subgroup forces the whole group to be a direct product. See A Sylow $p$ -subgroup is normal if and only if it is unique.

Facts & Assumptions

Given: The hypotheses and objects in the false claim.

[L1]

A Sylow $p$ -subgroup of a finite group is normal if and only if it is the unique Sylow $p$ -subgroup. (A Sylow $p$ -subgroup is normal if and only if it is unique).

[L2]

Let $p < q$ be primes. - If $p ∤ (q - 1)$ , every group of order $p q$ is cyclic. - If $p ∣ (q - 1)$ , there are exactly two isomorphism classes of groups of order $p q$ : the cyclic group $C_{p q}$ and one nonabelian semidirect product $C_{q} ⋊ C_{p}$ . (Classification of groups of order $p q$ for primes $p < q$ ).

[L3]

The canonical factors of $N ⋊_{α} H$ form an internal direct product if and only if $α_{h} = {id}_{N}$ for every $h \in H$ . In that case $N ⋊_{α} H$ is the external direct product $N \times H$ . (The canonical semidirect decomposition is an internal direct product if and only if the defining action is trivial).

Refutation

technique · direct

1.1

Let $C_{2}$ act on $C_{3}$ by inversion. The resulting semidirect product $C_{3} ⋊ C_{2}$ has order $6$ and is nonabelian.

L1L2L3givenalgebra

2.1

Its canonical $C_{3}$ is normal and therefore the unique Sylow $3$ -subgroup. If the product were direct, the two factors would commute, contradicting the inversion action. This proves the stated claim.

step 1.1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 96 results over 16 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

Keith Conrad, Consequences of the Sylow Theorems, Sections 1-5 (standard reference, not scraped)