Alternate characterization of cosets as equivalence classes

Let G be a group and H \leq G. Define a relation \sim on G by a \sim b if and only if b^{-1} a \in H. Prove that \sim is an equivalence relation and describe for each a \in G the equivalence class [a]. Use this to prove the following proposition.

Proposition: Let G be a group and H \leq G. Then (i) the set of left cosets of H is a partition of G and (ii) for all u,v \in G, uH = vH if and only if v^{-1} u \in H.

  1. \sim is an equivalence:
    1. Reflexive: x^{-1}x = 1 \in H, so that x \sim x for all x \in G.
    2. Symmetric: Suppose x \sim y. Then y^{-1} x \in H. Since H is a subgroup, (y^{-1}x)^{-1} = x^{-1} y \in H, and we have y \sim x.
    3. Transitive: Suppose x \sim y and y \sim z. Then y^{-1}x \in H and z^{-1}y \in H. Thus we have y^{-1}x = h_1 and z^{-1}y = h_2 for some h_1,h_2 \in H. Now y = xh_1^{-1}, and substituting yields z^{-1}xh_1^{-1} = h_2, hence z^{-1}x = h_2h_1. Thus x \sim z.

    So \sim is an equivalence.

  2. Let x \in G and suppose y \sim x. Then x^{-1}y = h for some h \in H, hence y = xh. Thus y \in xH; so [x] \subseteq xH. Now suppose y \in xH. Then y = xh for some h \in H, hence x^{-1}y \in H and we have y \sim x, so that xH \subseteq [x]. Thus the \sim-equivalence classes of G are precisely the left cosets of H.
  3. Proof of Proposition 4: The left cosets of H form the equivalence classes of a relation \sim on G defined by x \sim y if and only if y^{-1}x \in H; the second conclusion of Proposition 4 follows trivially.
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