Math 551, Assignment 2

Due Thursday, Sept. 28 in class

1. An action of a group G on a set Ω is said to be doubly transitive, or 2-transitive, if and only if for every (α, β) and (γ, δ) ∈ Ω × Ω such that α ≠ β and γ ≠ δ, there exists g ∈ G such that gα = γ and gβ = δ.

a) Show that if |Ω| > 2, then G is doubly transitive on Ω if and only if G is transitive on Ω and also G_{α} is transitive on Ω − {α} for some α ∈ Ω.

b) Let n be an integer such that n > 1. Show that the natural action of SL_{n}(C) on C^{n} − {0} (i.e. the set of all nonzero n × 1 column vectors) is transitive but not doubly transitive, while the natural action of SL_{n}(C) on the set of 1-dimensional subspaces of C^{n} is doubly-transitive.

2. The quaternion group Q_{8} can be defined as the subgroup ⟨ [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] , [ \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ] ⟩ of

SL_{2}(C). Demonstrate the following: a) |Q_{8}| = 8; b) Q_{8} contains a unique element of order 2; c) Every subgroup of Q_{8} other than Q_{8} itself is cyclic; d) Q_{8} ≅ D̸_{8}.

3. Let G be a group of order 264. What can you say about the number of elements of order 11 in G? Can these elements all be conjugate in G?

4. Show that if p is a prime divisor of the finite group G, then the number of subgroups of G of order p is congruent to 1 mod p. (Hint. Begin by adapting the proof of the corresponding part of Sylow’s Theorem.)

5. Let G = Σ_{4}. Write down representatives of the conjugacy classes of G (i.e. exactly one element from each conjugacy class). For each representative g which you have written, compute |C_{G}(g)| and give generators of C_{G}(g). Do the same for A_{4}. (The counting principle is useful here!)

6. a) Let H be any subgroup of Σ_{n}. Show that |H : H ∩ A_{n}| = 1 or 2.

b) Let g ∈ A_{n}. Show that each Σ_{n}-conjugate of g (i.e. each permutation of the same cycle shape as g) is already conjugate to g \underline{by an element of A_{n}} if and only if g centralizes some odd permutation in Σ_{n}.

c) Give an example of an element g ∈ A_{15} for which the conditions in b) are false.

7. Suppose that g, h ∈ Σ_{n} and gh = hg.

a) Show that if Ω_{1} is an orbit of ⟨g⟩ on {1, 2, …, n}, then hΩ_{1} is also such an orbit.

b) Express the orbits of C_{Σ_{n}}(g) in terms of the cycle decomposition of g.