Question

Given a group G with a subgroup H, define a binary relation on G by a ∼ b if and only if ba^(-1)∈ H.

(a) (5 points) Prove that ∼ is an equivalence relation.

(b) (5 points) For each a ∈ G denote by [a] the equivalence class of a and prove that [a] = Ha = {ha | h ∈ H}. A set of the form Ha, for some a ∈ G, is called a right coset of H in G.

(c) (5 points) Let a ∈ G. For all g ∈ G prove that Hg = Ha if and only if g ∈ Ha. Hint: two elements are equivalent if and only if their equivalence classes coincide.

(d) (5 points) Prove that the map ρa : H → Ha given by ρa(h) = ha, h ∈ H, is a bijection.

Answer 1