Question

Let G be a group with the binary operation of juxtaposition and identity e. Let H be a subgroup of G.

(a) (4 points) Prove that a binary relation on G defined by a ∼ b if and only if a−1b ∈ H, is an equivalence.

(b) (3 points) For all a ∈ G, denote by [a] the equivalence class of a with respect to ∼ . Prove that [a] = {ah|h ∈ H}. We write [a] = aH and say that aH is a left coset of H in G. Denote by π : G → G/ ∼ the quotient map of ∼ . What is the value of π(a)?

(c) (3 points) Prove that the map λa : H → aH given by λa(h) = ah is one-to-one and onto. If H is finite, what can you say about the cardinalities |H| and |aH|?

(d) (4 points) (Lagrange’s Theorem) If G is a finite group then |H| divides |G|. The quotient [G : H] = |G| is called the index of H in G. What is the meaning of the index? Hint: the left

|H|
cosets of H in G form a partition of G.

(e) (1 point) Let K be a subgroup of G. Denote by ◃▹ the equivalence relation on G given by a ◃▹ b if and only if a−1b ∈ K, let σ : G → G/ ◃▹ be the quotient map of ◃▹ . What is the value of σ(a)?

(f) (1 point) Prove that if K ⊆ H then ◃▹ is finer than ∼ .

(g) (4 points) Suppose K ⊆ H and denote by g : G/ ◃▹−→ G/ ∼ the unique map satisfying π = gσ, see Corollary 8 of the file “Finer Equivalences and Lifting Maps.” For all a ∈ G, what is the value of g(aK)?

Answer 1