Higher Algebra I — Fall 2005

Gerald Hoehn

Problem sheet 4 September 15, 2005

Problem 1: Let G be a finite group acting on a finite set X. For g ∈ G, let X^{g} be the fixpoints of g, i.e., X^{g} = {x ∈ X | gx = x}. Proof Burnside’s formula

|X/G| = \frac{1}{|G|} ∑_{g∈G} |X^{g}|

for the number of G-orbits. (Hint: Count the subset F = {(g, x) | gx = x} ⊂ G × X in two different ways.)

Problem 2: Let N be a normal subgroup of a finite group G. Let S ⊂ G be a conjugacy class of elements in G, and assume that S ⊂ N. Prove that S is a union of n conjugacy classes in N, all having the same cardinality, where n equals the index [G : N.Cent(x)] of the group generated by N and the centralizer Cent(x) in G of any element x ∈ S.

Problem 3: Let σ ∈ S_{n} such that in the cycle decompostion of σ there are λ_{i} cycles of length i for i = 1, …, n.

(a) Describe the centralizer of σ in S_{n} and determine its order.

(b) Describe the normalizer of the cyclic group ⟨σ⟩ generated by σ in S_{n} and determine its order.

(c) Determine the number of pairs of commuting elements in S_{n}, i.e., the number of pairs (σ, τ) with σ, τ ∈ S_{n} and στ = τσ.

Problem 4: Proof that A_{n}, the alternating group of degree n, is simple for n ≥ 5. (Hint: Use that A_{n} is generated for n ≥ 5 by 3-cycles. Let σ ∈ N \ {e} be an element of a normal subgroup N of A_{n} with a maximal number of fixpoints. Show that σ is conjugated to a 3-cycle or an element with more fixpoints. Distinguish the two cases that all orbits of ⟨σ⟩ have size 1 and 2 or that there is at least one orbit of size 3 or larger.)

Problem 5^{∗} (the conjugacy classes of A_{n}): Let T ⊂ S_{n} be a conjugacy class in S_{n}, i.e., the set of all permutations σ ∈ S_{n} which have the same cycle type and assume that T ⊂ A_{n}. Show that each such conjugacy class T is either also a conjugacy class in A_{n} or decomposes in two conjugacy class of A_{n} of the same size; the second case happens if and only if the cycle type of an element of T consists of distinct odd integers.