Reference no: EM132262837

Math Questions -

Q1. Let G be a group, and let F be the set of all functions from the real numbers to G. So, F = {R : R → G}. For f, g ∈ F, let f ? g, be defined by (f ? g)(x) = f(x)g(x).

(a) Prove that F is a group under ?. (Hint: this is not a subgroup of anything that we have seen).

(b) Let H ≤ G. Let F_H = {f ∈ F|f(x) ∈ H for all x ∈ R}. Prove that F_H ≤ F.

(c) Now say H G. Is F_H F? Prove or disprove.

Q2. Let p, q, r be distinct primes and let n = pqr.

(a) List all the subgroups of Z_n without repetition. Which are contained in which? (A labeled picture is fine here).

(b) If p, q, r are all odd, what is |([2pq])|?

Q3. Let G be a group and let K = {g ∈ G|g = x² for some x ∈ G}.

(a) Prove that if G is abelian, K ≤ G.

(b) If G = S₃, is K ≤ G?

(c) If n ≥ 1 is odd, and G = Z_n what is K? Prove your answer.

Q4. Let G be a group and let S = {H|H ≤ G}. Define ∼ on S by H₁ ∼ H₂ if there is a g ∈ G with H₁ = g(H₂)g^-1.

(a) Prove that ∼ is an equivalence relation on S.

(b) If N G what is [N]?

(c) Find all x ∈ S such that x ∼ G?

Q5. Consider the subgroup H = {e, (12)(34), (13)(24), (14)(23)} ≤ S₄.

(a) Show that H is not cyclic.

(b) We have shown that H S₄. Is G/H is abelian? Prove your answer.

(c) What is |(123)H| in G/H?