How 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.
The coset set and the index of a subgroup
Definition
Let . The left coset set is
By The left cosets of a subgroup partition the group, its elements are exactly the blocks of the coset partition of . The index of in is
when is finite, with finite cardinality as in The cardinality of a finite set. If is not finite, write . Here is a symbol, not a natural number, and no arithmetic with it is defined.
The right coset set has the same finite or infinite size because Inversion induces a bijection from left cosets to right cosets gives an explicit bijection between the two coset sets. Thus the index does not depend on choosing left rather than right cosets.
Depends on
Used by
- [G:H]=1 if and only if H=G Corollary
- Every subgroup of index p in a finite p-group is normal Corollary
- For K≤ H≤ G with G finite, [G:K]=[G:H][H:K] Corollary
- If [G:N] is finite then |G/N|=[G:N]; for finite G this equals |G|/|N| Corollary
- Orbit-stabiliser cardinality: |G· x|=[G:Gₓ] whenever either side is finite, and |G|=|Gₓ| |G· x| for finite G Corollary
- 2ℤ has index 2 in ℤ and is nevertheless equinumerous with ℤ Counterexample
- The quotient group G/N and coset product (gN)(hN)=ghN Definition
- [G:G]=1 and, for finite G, [G:{e}]=|G| Example
- For n≥1, the cosets of nℤ are the n congruence classes modulo n Example
- The three-cycle subgroup of Sym({1,2,3}) is normal and its quotient has two elements Example
- In a finite group, the subgroup, every coset and the set of cosets are finite Lemma
- Every subgroup of index two is normal Theorem
- If [G:H]=n<∞, then Core_G(H) is normal in G, [G:Core_G(H)]∣ n!, and only finitely many subgroups contain H Theorem
- Lagrange's theorem: |G|=[G:H]|H| for every subgroup H of a finite group G Theorem
- Left multiplication on G/H is transitive, has stabiliser H at H, and has kernel Core_G(H) Theorem
- The conjugates of a proper subgroup do not cover a finite group Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 46 results over 16 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, §6.2: Lagrange's Theorem (standard reference, not scraped)