CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-01

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A left coset that is not the corresponding right coset in $Sym ({1, 2, 3})$

Statement refuted

For every subgroup $H \leq G$ and every $g \in G$ , the corresponding cosets $g H$ and $H g$ are equal.

Facts & Assumptions

Given: The group $S_{3} = Sym ({1, 2, 3})$ , the subgroup $H = {e, (12)}$ , and $g = (123)$ .

[F1]

Cycle products are composites with the rightmost permutation acting first (The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition, $Sym (X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements).

[F2]

The sets $g H = {g h : h \in H}$ and $H g = {h g : h \in H}$ are the left and right cosets (Left and right cosets $g H$ and $H g$ of a subgroup).

[F3]

Under rightmost-first composition, if $τ = (12)$ then $e e = e$ , $e τ = τ e = τ$ and $τ^{2} = e$ ; hence $e^{- 1} = e$ , $τ^{- 1} = τ$ , and $H = {e, τ}$ contains the identity and is closed under products and inverses, so it is a subgroup (The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition, $Sym (X)$ is a group under composition, and it is non-abelian whenever $X$ has at least three distinct elements, Subgroup).

Counterexample

technique · direct

1.1

Direct composition gives $(123) (12) = (13)$ and $(12) (123) = (23)$ .

2.1

Therefore $g H = {(123), (13)}$ while $H g = {(123), (23)}$ .

step 1.1F2F3

3.1

Since $(13) \neq (23)$ , the left and right cosets are unequal, refuting the statement.

step 2.1∎

Depends on

Used by

Every left coset of a subgroup is itself a subgroup False statement

Dependency tree · two levels

10 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

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)

A left coset that is not the corresponding right coset in Sym⁡({1,2,3})Sym({1,2,3})