Alphabeta MathHow statement and proof provenance work
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.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
iff , and iff
Statement
Let and let . Then
and
The corresponding right-coset criterion is if and only if .
Facts & Assumptions
Given: A group , a subgroup , and elements .
The left coset is , and the right coset is (Left and right cosets and of a subgroup).
A subgroup contains the identity and is closed under products and inverses (Subgroup).
In a group, and (In a group , and , the order of the last product being essential).
Left and right cancellation hold in every group (Cancellation in a group: or forces ; equivalently left and right translation by are bijections of , so and each have exactly one solution, Group and abelian group).
Proof
If , write with ; then . Conversely, if , then .
Suppose . If , write ; subgroup closure gives , so . Thus .
If , then , so step 1.1 gives .
Since , the same argument with interchanged gives . Hence .
Finally, is equivalent, after taking inverses elementwise, to ; by the left-coset criterion this holds exactly when .
Depends on
- Left and right cosets $gH$ and $Hg$ of a subgroup
- Subgroup
- Group and abelian group
- In a group $e^{-1} = e$, $(g^{-1})^{-1} = g$ and $(gh)^{-1} = h^{-1}g^{-1}$, the order of the last product being essential
- Cancellation in a group: $gx = gy$ or $xg = yg$ forces $x = y$; equivalently left and right translation by $g$ are bijections of $G$, so $gx = h$ and $xg = h$ each have exactly one solution
Used by
- [G:H]=1 if and only if H=G Corollary
- A nonnormal two-element subgroup of Sym({1,2,3}) makes coset multiplication depend on representatives Counterexample
- If G/Z(G) is cyclic, then G is abelian Lemma
- Inversion induces a bijection gH↦ Hg⁻¹ from left cosets to right cosets Lemma
- The left cosets of a subgroup partition the group Lemma
- Two elements have the same image under a homomorphism if and only if they lie in the same coset of its kernel Lemma
- In ⟨ X∣ R⟩, the words u and v represent the same element if and only if u⁻¹v∈⟨⟨ R⟩⟩ Proposition
- The canonical map M→ M/N is a surjective module homomorphism with kernel N; thus every submodule is a kernel Proposition
- The canonical projection R→ R/I is a surjective ring homomorphism with kernel I Proposition
- Coset multiplication (gH)(hH)=ghH is well defined if and only if H is normal Theorem
- Every subgroup of index two is normal Theorem
- G/N is abelian if and only if [G,G]⊆ N Theorem
- Left multiplication on G/H is transitive, has stabiliser H at H, and has kernel Core_G(H) Theorem
- Multiplication of additive cosets is well defined if and only if the additive subgroup is a two-sided ideal Theorem
- Orbit-stabiliser: G/Gₓ→ G· x, gGₓ↦ g· x, is a well-defined bijection Theorem
- The quotient action is well defined and makes M/N a module Theorem
Dependency tree · two levels
11 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
- UCL lecture notes, Cosets and Lagrange's theorem (standard reference, not scraped)
- Thomas W. Judson, Abstract Algebra: Theory and Applications, §6.1: Cosets (standard reference, not scraped)