Group Theory: Math30038, Sheet 5

GCS

1. Let G = S_{n} be the symmetric group on {1, 2, …, n}, so |G| = n! and the elements of G are the permutations of {1, 2, …, n}.

(a) Suppose that (a_{1}, a_{2}, …, a_{t}) ∈ G is a cycle, and that g ∈ G. Show that g^{−1}(a_{1}, a_{2}, …, a_{t})g = (a_{1}g, a_{2}g, …, a_{t}g).

(b) Each element of G can be expressed as a product of disjoint cycles (elements of G are disjoint if their supports are disjoint). Show that the number of conjugacy classes in S_{n} is the number of ways of writing n as an ascending sum a_{1} + a_{2} + ⋯ + a_{t} of positive integers a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{t}. Thus there are 3 conjugacy classes in S_{3} because 3 can be written as an ascending sum in three ways: 3, 1 + 2, 1 + 1 + 1. Also in S_{4} there are 5 conjugacy classes because 4 is 4, 1 + 3, 1 + 1 + 2, 2 + 2 and 1 + 1 + 1 + 1.

(c) Determine the number of conjugacy classes in S_{5}, and the size of each conjugacy class, and describe the centralizer in G of a chosen representative of each conjugacy class.

2. Let G = S_{n}. Let x = (1, 2, …, n) ∈ G. Prove that C_{G}(x) = ⟨x⟩.

3. Show that in S_{4} there is a non-identity element y such that C_{G}(y) ≠ ⟨y⟩.

4. Suppose that G is a finite group. Show that the number of elements in each conjugacy class of G must divide G.

5. Let G be a group with a subgroup H such that g^{−1}Hg ⊆ H for every g ∈ G. Prove that g^{−1}Hg = H for every g ∈ G.

6. (Challenge) Does there exist a group G containing an element g and a subgroup H such that g^{−1}Hg ⊆ H but g^{−1}Hg ≠ H.

7. Suppose that G is a group and that H ≤ G. Choose g ∈ G. Prove that g^{−1}Hg ≤ G.

8. Let G be a finite group of order n which has t conjugacy classes. Elements x and y are each selected uniformly at random from G. What is the probability that x and y commute? Does this make sense for abelian groups?

9. Show that a finite group with exactly two conjugacy classes must have two elements.

10. Let G be a containing H a subgroup of finite index. Let S = {x^{−1}Hx | x ∈ G}. Let G act on S by conjugation so if K ∈ S then K ·g = g^{−1}Kg. Verify that this is a group action, and deduce that |S| = |G : N_{G}(H)| where N_{G}(H) = {g ∈ G | gH = Hg}. Deduce that |S| is finite and divides |G : H|.