Math 551, Assignment 3

Due Thursday, Oct. 5 in class

1. Let G be a group and S a subset of G such that G = ⟨S⟩. Suppose that T is a nonempty subset of G such that sT ⊆ T for all s ∈ S. Show that T = G.

2. For all positive integers i, j, let m_{ij} = 1 or 3 if |i − j| = 0 or 1, respectively, and let m_{ij} = 2 otherwise. Show that for any n ≥ 1, gp⟨t_{1}, …, t_{n} | (t_{i}t_{j})^{m_{ij}} = 1⟩ ≅ Σ_{n+1}. (Hint. Let G be this group and H the subgroup generated by t_{1}, …, t_{n−1}. Show that t_{n}H ∪ (t_{n−1}t_{n})H ∪ (t_{n−2}t_{n−1}t_{n})H ∪ ⋯ ∪ (t_{1} ⋯ t_{n})H = G, and bound |H| by induction.)

3. Find all maximal subgroups of Σ_{4} (i.e. those subgroups H < Σ_{4} for which there are no subgroups K such that H < K < Σ_{4}). Sylow’s Theorem is useful!

4. Suppose that H ≤ G and |G : H| is finite. Show that there exists K ◁ G such that K ≤ H and |G : K| is finite.

5. Let ϕ : G → H be a homomorphism. Let X and Y be subgroups of G. Prove that if ker ϕ ≤ X, then ϕ(X ∩ Y) = ϕ(X) ∩ ϕ(Y). Show however that if the assumption on ker ϕ is removed, the statement becomes false in general.

6. Let ϕ : G → H be a homomorphism, and let x ∈ G. Show that if x has finite order, then so does ϕ(x), and the order of ϕ(x) divides that of x.

7. Let H ◁ G and let π_{H} : G → G/H be the projection. Let p be a prime. Show that if P is a Sylow p-subgroup of G, then π_{H}(P) and P ∩ H are Sylow p-subgroups of G/H and H, respectively. Show by example that P ∩ H is not necessarily a Sylow p-subgroup of H if the normality assumption on H is omitted.

8. Let K ≤ H ≤ G and assume that K ◁ H. Let A be a subgroup of G. Show that A ∩ K ◁ A ∩ H, and A ∩ H/A ∩ K is isomorphic to a subgroup of H/K.

9. Suppose that you are given presentations for two groups, say X_{1} = gp⟨S_{1} | R_{1}⟩ and X_{2} = gp⟨S_{2} | R_{2}⟩. Give presentations for

a) the free product X_{1} ∗ X_{2} and

b) the direct product X_{1} × X_{2}.

Prove that your answers are correct.