TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-17

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Every group of order $p^{2} q$ for distinct primes has a normal Sylow subgroup

Statement

Every group of order $p^{2} q$ for distinct primes has a normal Sylow 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 Statement.

[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]

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).

Proof

technique · direct

1.1L1L2givenalgebra

If $p > q$ , the restrictions $n_{p} ∣ q$ and $n_{p} \equiv 1 (m o d p)$ force $n_{p} = 1$ .

2.1step 1.1givenalgebra

If $q > p$ , then $n_{q} \in {1, p^{2}}$ ; the second value forces $q ∣ (p - 1) (p + 1)$ , hence the sole exceptional pair $(p, q) = (2, 3)$ .

3.1step 2.1givenalgebra

For order $12$ , if the four Sylow $3$ -subgroups are nonnormal, their eight nonidentity elements leave exactly four elements, so every Sylow $2$ -subgroup is that same four-element complement and is normal.

4.1step 1.1step 2.1step 3.1L2givenalgebra∎

The two orderings exhaust the hypothesis, since $p$ and $q$ are distinct: step 1.1 settles $p > q$ with a normal Sylow $p$ -subgroup, and step 2.1 settles $q > p$ with a normal Sylow $q$ -subgroup except at $(p, q) = (2, 3)$ , which step 3.1 settles with a normal Sylow $2$ -subgroup. Every case therefore produces a normal Sylow subgroup, and [L2] turns each uniqueness count $n = 1$ into normality. This proves the stated claim.

Depends on

Used by

No group of order p²q for distinct primes is simple Corollary

Dependency tree · two levels

8 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)

Every group of order p2qp2q for distinct primes has a normal Sylow subgroup