Math 551, Assignment 3, due Wednesday, October 16 in class

As usual, G is a group.

1. Prove the following two statements using group actions.

a) Suppose that H ≤ G and |G : H| = n < ∞. Show that there is a subgroup K ≤ H such that K ◁ G and |G : K| divides n!, and in particular |G : K| < ∞.

b) Suppose that H_{1} ≤ G and H_{2} ≤ G are two subgroups of finite index in G. Show that H_{1} ∩ H_{2} has finite index in G. (Hint. Use an action on the disjoint union (G/H_{1}) ∪ (G/H_{2}).)

2. Let Ω be an infinite set and define

Σ_{Ω}^{o} = {σ ∈ Σ_{Ω} | σ(ω) = ω for all but finitely many ω ∈ Ω}.

a) Show that Σ_{Ω}^{o} ◁ Σ_{Ω}.

b) Make a reasonable definition of A_{Ω}^{o}, and show that A_{Ω}^{o} is simple.

3. Suppose that G = ⟨S⟩, and let X be a subset of G. Show that X = G if and only if (S ∪ S^{−1})X ⊆ X, where S^{−1} = {s^{−1} | s ∈ S}.

4. Let G = gp⟨s_{1}, …, s_{n} |R⟩, where R is the set of all relators of the form

\begin{matrix}s_{i}^{2}, & 1 ≤ i ≤ n \\ (s_{i}s_{j})^{2}, & 1 ≤ i < j ≤ n, j > i + 1 \\ (s_{i}s_{i+1})^{3}, & 1 ≤ i < n \end{matrix}

Show that G ≅ Σ_{n+1}. (Hints. Map s_{i} ↦ (i i + 1) ∈ Σ_{n+1}. Also, let H be the subgroup of G generated by s_{1}, …, s_{n−1}, and show that G = H ∪ s_{n}H ∪ s_{n−1}s_{n}H ∪ ⋯ ∪ s_{1}s_{2} ⋯ s_{n}H.)

5. Each of the following statements is false. Give a counterexample in each case. Also alter the underlined part of the statement in as reasonable way as possible so that the statement becomes true; then prove the altered statement.

a) If ϕ : G → H is any homomorphism and L and M are \underline{any subgroups} \underline{of G}, then ϕ(L ∩ M) = ϕ(L) ∩ ϕ(M).

b) If K char G and K ≤ L ≤ G, then L char G \underline{if and only if} L/K char G/K.

c) If P ∈ Syl_{p}(G) and H is \underline{any subgroup of G}, then P ∩ H ∈ Syl_{p}(H).

d) If G possesses a composition series and H is \underline{any subgroup of G}, then H possesses a composition series.

e) If A, B and C are subgroups of G such that A ◁ B, then A∩C ◁ B∩C and \underline{B ∩ C/A ∩ C ≅ B/A}.

6. If k is any field and n is any natural number, then we define Z = {cI | c ∈ k^{×}}, P GL_{n}(k) = GL_{n}(k)/Z and P SL_{n}(k) = SL_{n}(k)/Z ∩ SL_{n}(k). Show that P GL_{n}(k) has a normal subgroup H ≅ P SL_{n}(k), and that P GL_{n}(k)/H ≅ k^{×}/(k^{×})^{n}.

7. Suppose that G is a finite solvable group. Show that if x_{1}, …, x_{r} ∈ G are elements of orders p_{1}^{a_{1}}, …, p_{r}^{a_{r}}, respectively, where p_{1}, …, p_{r} are distinct primes and a_{1}, …, a_{r} are nonnegative integers, and if x_{1}x_{2} ⋯ x_{r} = 1, then a_{1} = a_{2} = ⋯ = a_{r} = 0. (Hint. Use induction. Remark. It is also true that any finite group with this property is solvable, but this is a very deep theorem.)

8. Show that up to isomorphism, there are exactly three nonabelian groups of order 12: A_{4}, Z_{2} × Σ_{3} and gp⟨x, y | x^{4}, y^{3}, xyx^{−1}y⟩.