False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-17

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False statement: every group of order $42$ has a normal Sylow $2$ -subgroup

Statement

False claim: every group of order $42$ has a normal Sylow $2$ -subgroup. See Sylow III: $n_{p} \equiv 1 (m o d p)$ and $n_{p} ∣ m$ when $∣ G ∣ = p^{a} m$ with $p ∤ m$ .

Facts & Assumptions

Given: The hypotheses and objects in the false claim.

[L1]

Let $∣ G ∣ = p^{a} m$ with $p ∤ m$ . Then the number of Sylow $p$ -subgroups satisfies $n_{p} (G) \equiv 1 (m o d p), n_{p} (G) ∣ m .$ . (Sylow III: $n_{p} \equiv 1 (m o d p)$ and $n_{p} ∣ m$ when $∣ G ∣ = p^{a} m$ with $p ∤ m$ ).

[L2]

Let $N$ and $H$ be groups (def-group), and let $α : H \to Aut (N)$ be an action by automorphisms (def-action-by-automorphisms). The external semidirect product $N ⋊_{α} H$ is the set $N \times H$ with multiplication. ( The external semidirect product $N ⋊_{α} H$ ).

[L3]

For every prime $p$ , the operations of addition and multiplication on $Z / p$ make it a field (def-field). (For every prime $p$ , the two operations on $Z / p$ make it a field).

Refutation

technique · direct

1.1L1L2L3givenalgebra

We construct the affine group $F_{7} ⋊ F_{7}^{\times}$ of order $42$ .

2.1step 1.1givenalgebra

Its seven involutions $x \mapsto - x + b$ generate seven Sylow $2$ -subgroups, so none is normal.

3.1step 2.1L1givenalgebra∎

The translations $x \mapsto x + b$ form a subgroup of order $7$ , and it is normal because $(x \mapsto a x + c)$ conjugates $x \mapsto x + b$ to $x \mapsto x + a b$ , again a translation; so the group does have a normal Sylow $7$ -subgroup, and it is only the Sylow $2$ -subgroups that fail to be normal. The count $n_{2} = 7$ of step 2.1 is consistent with [L1], since $7 \equiv 1 (m o d 2)$ and $7 ∣ 21$ . This proves the stated claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

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

False statement: every group of order 4242 has a normal Sylow 22-subgroup