CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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The product set $HK$ of two subgroups need not be a subgroup

Statement refuted

For any two subgroups $H,K\le G$ , the product set $HK=\{hk:h\in H,\ k\in K\}$ is a subgroup of $G$ .

Facts & Assumptions

Given: The group $S_3=\operatorname{Sym}(\{1,2,3\})$ , and the subgroups $H=\{e,(12)\}$ and $K=\{e,(23)\}$ .

[F1]

$S_3$ is a group under rightmost-first composition, and $H,K$ are subgroups because each transposition squares to $e$ (The symmetric group $\operatorname{Sym}(X)$ : the bijections of a set $X$ under composition, $\operatorname{Sym}(X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements, Subgroup).

[L1]

If $H$ is a subgroup of a finite group $G$ , then under the canonical embedding $\iota:\mathbb N\to\mathbb Z$ one has $\iota(|H|)\mid\iota(|G|)$ (Lagrange's theorem: $|G|=[G:H]|H|$ for every subgroup $H$ of a finite group $G$ , The order $|G|$ of a finite group and the order $\operatorname{ord}(g)$ of an element, with $\operatorname{ord}(g) = \infty$ when no positive power of $g$ is the identity).

Counterexample

technique · direct

1.1

The six elements of $S_3$ are $e,(12),(13),(23),(123),(132)$ , so $|S_3|=6$ .

1.2

Direct multiplication gives $HK=\{e,(12),(23),(12)(23)\}=\{e,(12),(23),(123)\}$ , a set of four distinct elements.

2.1

If $HK$ were a subgroup, [L1] would force $\iota(4)\mid\iota(6)$ in $\mathbb Z$ , that is $4\mid6$ , which is false. Hence $HK$ is not a subgroup.

step 1.1step 1.2L1∎

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: 75 results over 19 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

Thomas W. Judson, Abstract Algebra: Theory and Applications, Cosets and Lagrange's Theorem (standard reference, not scraped)
Thomas W. Judson, Abstract Algebra: Theory and Applications, §6.1: Cosets (standard reference, not scraped)

The product set HKHKHK of two subgroups need not be a subgroup