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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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

Statement refuted

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

Facts & Assumptions

Given: The group $S_3=\operatorname{Sym}(\{1,2,3\})$ , the subgroup $H=\{e,(12)\}$ , and $g=(123)$ .

[F1]

Cycle products are composites with the rightmost permutation acting first (The symmetric group $\operatorname{Sym}(X)$ : 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).

[F2]

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

[F3]

Under rightmost-first composition, if $\tau=(12)$ then $ee=e$ , $e\tau=\tau e=\tau$ and $\tau^2=e$ ; hence $e^{-1}=e$ , $\tau^{-1}=\tau$ , and $H=\{e,\tau\}$ contains the identity and is closed under products and inverses, so it is a subgroup (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).

Counterexample

technique · direct

1.1

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

2.1

Therefore $gH=\{(123),(13)\}$ while $Hg=\{(123),(23)\}$ .

step 1.1F2F3

3.1

Since $(13)\ne(23)$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 11 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)

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