Mathematics 666
Homework (due Oct. 9)
A. Hulpke
21) Determine the character table (i.e. the characters of the irreducible representations) of the
dihedral group of order 8.
22) Determine the character table of S4 . Hints:
a) S4 possesses a normal subgroup V ⊲ S4 of order 4 with S4 /V ≅ S3 . Thus every S3 representation
becomes an S4 representation by composition with the natural homomorphism (this is sometimes
called inflation).
b) Decompose the character of the natural permutation representation of S4 .
c) Use the character of the regular representation of S4 .
23) Let G be a finite group and χ an irreducible character of G. (We know that the values of χ lie
in Q(ε).)
Let σ ∈ Gal(Q(ε)/Q) be a field automorphism. We define χ σ by
χ σ ∶ G → C, g ↦ (χ(g))σ
Then χ σ is a character. (Proof: χ stems from a representation δ, defineable over a finite degree
extension F of Q(ε). Extend σ to an automorphism of F, then the representation δ σ affords χ σ .)
a) Let g ∈ G with ∣g∣ = m. Show that χ(g) ∈ Q for all χ ∈ Irr(G) if and only if for every l with
gcd(l , m) = 1 the elements g and g l are in the same conjugacy class.
b) Show that the character table of the symmetric group S n contains only rational values. (As character values also are algebraic integers, this implies that the entries are in fact integers.)
24) Let G = H × K be the direct product of the finite groups H and K. For χ ∈ Irr(H), ξ ∈ Irr(K)
we define χ × ξ by (χ × ξ)(h, k) ∶= χ(h) ⋅ ξ(k). Show:
a) χ × ξ is an irreducible character of G. (You need to show first that it is the character of a representation, and then that it is irreducible)
b) Every irreducible character θ of G is of the form θ = χ × ξ for suitable χ ∈ Irr(H), ξ ∈ Irr(K).
25) Let V be a CG-module with corresponding character χ. According to problem 18, we can
decompose V ⊗V = WS ⊕WA. Let χ S (respectively χ A) the characters that belong to WS (respectively
WA). Show that
χ S (g) =
1
(χ(g)2 + χ(g 2 ))
2
and
χ A(g) =
1
(χ(g)2 − χ(g 2 ))
2