Skip to content
Axiomatic Wiki
Lemma·L29

The coset equivalence relation

Sharing a coset is an equivalence, and the classes are the cosets.

Let be a subgroup and define the relation on by Then (i) is an equivalence relation on , and (ii) its classes are the left cosets: for every .
In words
For a subgroup H of a group G, call two group elements a and b related exactly when the inverse of a times b lands in the subgroup. This relation is reflexive, symmetric and transitive, and the equivalence class is exactly the left coset, for every element a.
Rests onF02F05F09F13F14A01A02A06
Never needed: F03 · F04 · F06 · F08 · F10 · F11 · F12 · A03 · A04 · A05 · A07 · A08 (computed from the citation graph, not asserted).

Proof

  1. 1
    is a set of pairs by Separation inside , hence a relation on (D012, D011).
  2. 2
    Reflexive: by L28 (ii) and the identity clause of D039.
  3. 3
    Symmetric: let . By closure under inverses (D039), , and by L28 (v) and (iv): So , i.e. .
  4. 4
    Transitive: let and . Closure under the operation (D039) puts their product in , and associativity with the inverse and identity laws (D038 (G1), L28) computes So , i.e. .
  5. 5
    (ii), classes are cosets: fix . If , i.e. (D021), then by (G1), (G2, G3) and D040. Conversely if , say with , then so . Extensionality gives .
∎

Remarks

Congruence modulo a subgroup: for the additive group of integers and the subgroup of multiples of , this relation is precisely congruence mod . Feeding (i) into T04 shows the left cosets partition ; with the uniform size from L30, the partition counts itself into Lagrange's theorem.
Wikipedia ↗QuotientGroup.leftRel ↗

Used by

T13
Propose an edit1 published revision