Dummit and Foote
Abstract Algebra, section 18, exercises 11, 12, 14-24:
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Let
G be a finite group of order
n.
11. Let
χ be an irreducible character of
G. Prove that for every
z∈Z(G) we have
χ(z) = ζχ(1) for some root of unity
ζ.
12. Let
ψ be the character of some representation
φ of
G, and let
g∈G. Prove (a)
ψ(g) = ψ(1) if and only if
g∈\text{ker }φ, and (b)
|ψ(g)| = ψ(1) implies
g ∈ Z(G) if
φ is faithful.
In the exercises that follow, let
F = \overline{ℚ} be the field of algebraic numbers.
14. Prove that a representation
φ of
G over
F is irreducible over
F if and only if it is irreducible when considered as a representation over
ℂ. Deduce that the group characters of
G when considered over
F are the same as when considered over
ℂ, and that every complex representation of
G is equivalent to a representation over
F.
16-17. Let
σ be an automorphism of the field extension
F/ℚ. If
φ is a representation of
G over
F with character
ψ, let
φ^σ be the mapping given by applying the automorphism
σ to each of the entries of
φ. Prove that
φ^σ is a representation with character
ψ^σ = σ∘ψ.
18. If
ψ is again a character of
G, let
ℚ(ψ) be the (finite) extension of
ℚ by each value
ψ(g)∈F. Prove that
ℚ(ψ)⊆ℚ(ζ_n), where
ζ_n is a primitive
n^\text{th} root of unity. Deduce that
ℚ(ψ) is a Galois extension of
ℚ with abelian Galois group.
19-24. For each
a relatively prime to
n, let
σ_a ∈ \text{Gal}(ℚ(ζ)/ℚ) ≅ (ℤ/nℤ)^× be the automorphism given by
ζ_n ↦ ζ_n^a. Show that
ψ^{σ_a}(g) = ψ(g^a). Prove that every character of a group has all integer values if and only if for all
g∈G, one has
g is conjugate to
g^k for all
k relatively prime to
|g|.
Proof: (11) Let
φ be the corresponding representation of
G on
V. Since
z commutes with every element of
G, it commutes with every element of
ℂG. Therefore, multiplication by
z is a
ℂG-module endomorphism on
V. Diagonalize
V with respect to
z, so that
φ(z) is given by a matrix with roots of unity on the diagonal and zeros elsewhere. Let
ζ be one such root. By Schur's lemma,
z - ζ is either an automorphism or an annihilator of
V. Since
z - ζ annihilates at least one basis element of
V (that from which
ζ was chosen from
φ(z)), we see it must be the latter, so that
φ(z) = ζI and
χ(z) = ζχ(1).
(12)(a-b) If
g ∈ \text{ker }φ, then clearly
ψ(g) = ψ(1). Conversely, if
ψ(g) = ψ(1), then after diagonalization with respect to
g, we may consider
g as a matrix with
ψ(1) roots of unity
ζ_i on the diagonal. If
ψ(g) = \sum ζ_i = ψ(1), then also
|ψ(g)| = |\sum ζ_i| = \sum |ζ_i| = ψ(1), so by the triangle inequality each
ζ_i is equal. From here we see (b), whereby
φ(g) commutes with every matrix in
φ(G)⊆\text{GL}(ℂ), then
φ(g)∈Z(φ(G)) implying
g∈Z(G) if
φ is faithful. Given the stronger condition of (a) that
ψ(g) = ψ(1), we in fact see
ζ_i = 1, so
g∈\text{ker }φ.
(14) Note that the computation of a character norm is independent of the field over which the matrices are considered, and irreducibility of a character/representation is equivalent to the norm being
1. Thus, a representation is irreducible over
F if and only if it is irreducible over
ℂ. Furthermore, there are
r irreducible characters over
F that are equivalent to their corresponding complex characters, where
r is the number of conjugacy classes of
G; this is also independent of the field considered, so this is the full set of irreducible complex characters of
G. Thus, the
F-characters and complex characters are the same. Finally, since every representation over
ℂ will entail a character identical to that of some representation over
F, this implies every representation over
ℂ is equivalent to one over
F.
(16-17) The action of
σ on
\text{GL}(F) is in fact an automorphism, so the composition
φ^σ is a homomorphism from
G into
\text{GL}(V) corresponding to which we have clearly have the character
ψ^σ.
(18) Examining Proposition 14 more closely, we see in fact that
ψ(g)∈ℚ(ζ_k) for each
g∈G with order
|g|=k. Since necessarily
k will divide
n, and
ℚ(ζ_k)⊆ℚ(ζ_n), we have
ℚ(ψ)⊆ℚ(ζ_n), in fact
ℚ(ψ)=ℚ(ζ_e) for the exponent
e of
φ(G). Thus it is a Galois extension with abelian Galois group.
(19-24) If we diagonalize with respect to
g, we see that
σ_a(ζ_n^k) = (ζ_n^k)^a for all
k, hence
\text{tr }φ^{σ_a}(g) = \text{tr }φ(g)^a = \text{tr }φ(g^a) = ψ(g^a). If
g is conjugate to
g^a for all
a relatively prime to
|g|, then
ψ^{σ_a} = ψ as characters for all
a relatively prime to
n, so
ψ(g) is within the fixed field of the Galois extension
ℚ(ζ_n)/ℚ, that is,
ℚ. By Proposition 14, character values are algebraic integers, so
ψ actually takes values in
ℤ.
Conversely, suppose
g isn't conjugate to
g^a for some
a relatively prime to
|g|. Let
n = p_1^{α_1}p_2^{α_2}...p_k^{α_k} and
|g| = p_1^{β_1}p_2^{β_2}...p_k^{β_k} be factorizations, necessarily with
β_i ≤ α_i. Reordering the primes if necessary, let
β_i = 0 for
i > m. By the Chinese Remainder Theorem, let
b be such that
b≡a \mod p_i^{α_i}
for
i≤m, and
b≡1 \mod p_i^{α_i}
for
i > m. Then
b ≡ a \mod |g| (so
g^a = g^b), and
b is relatively prime to
n. Then since the irreducible characters of
G form a basis for the space of class functions on
G, there is some irreducible character
χ such that
χ(g)≠χ(g^b)=χ^{σ_b}(g)=σ_b(χ(g)). Thus
χ(g) is not within the fixed field of
σ_b, hence not an integer.
~\square