Following the notations in Theorem 3.48, then

D∈H\G/H

ι(σ,(gU)_H∩^g_H^H)

=

∑

D∈H\G/H

ι(U_H∩^g_H,(gU)_H∩gH)_.

This is equivalent to ι(U_H, U_H) ≡ ι(σ, σ) =1 for g =1, i.e. σ is irreducible, and ι(U_H∩^gH,(gU)_H∩^g_H) =0 for all g6∈ H. 

We conclude this section by a few important corollaries, notably (4).

Corollary 3.49. Following the notations in Theorem 3.48, then

(1) If deg σ=1, then σ^Gis irreducible if and only if for all g6∈ H, σ(h) 6=

σ(ghg⁻¹)for some h∈ H∩^gH.

(2) Let H/G. Then σ^G is irreducible if and only if σ is irreducible and

gσ6∼=σ for all g6∈H.

(3) Let H/G and T(σ):= {g ∈G |^gσ ∼=σ} ⊃ H. If τ is an irreducible representation of T(σ)such that τ_H contains σ then τ^Gis irreducible.

(4) Let H/G. If ρ is an irreducible representation of G such that σ is an irreducible component of ρ_H, then ρ=τ^Gfor some irreducible represen-tation τ of T(σ).

The group T(σ)is called the inertia group of σ.

PROOF. (1) is obvious.

(2) follows from the facts that^gH= H and the H-action on gU is^gσ.

(3): let τ : T := T(σ) → GL(V). Since H/T, Clifford’s theorem (The-orem3.3) says that all irreducible components of τ_H, i.e. V, are conjugated to σ. Hence they are all isomorphic to σ by the definition of T.

To show that V^G is irreducible, let g 6∈ T and consider V and gV as F[T∩^gT]-modules. By the above result for V, gV is then isomorphic to a sum of copies of gU as F[H]-modules. The H-action on gU is given ^gσ which is not isomorphic to σ since g 6∈ T. Thus V and gV are disjoint as F[H]-modules. Since T∩^gT ⊃ H, they are also disjoint as F[T∩^gT] -modules. Thus the result follows from Theorem3.48.

(4) Take an irreducible component τ of ρT such that τH ⊃ σ. By (3), τ^G is irreducible. Hence ρ=τ^Gby Theorem3.44(Frobenius reciprocity). 

5. BRAUER’S THEOREM ON INDUCED CHARACTERS 53

5. Brauer’s theorem on induced characters

Throughout this section we assume F⊂_{C and}|G| <_∞.

Definition 3.50. (1) A group G is p-elementary if G= Z×P with Z cyclic, P a p-group and p- |^Z|. G is elementary if it is p-elementary for some p.

(2) G is p-quasi-elementary if there is a cyclic Z/G, with G/Z a p-group and p- |^Z|. Clearly p-elementary⇒p-quasi-elementary.

It is clear tat the subgroup Z specified in Definition3.50is unique. Also any subgroup of a p(-quasi)-elementary group is p(-quasi)-elementary.

Lemma 3.51. (i) G is p-quasi-elementary⇐⇒G= AP with A/G being cyclic and P is a p-group. (ii) A p-quasi-elementary group is p-elementary⇐⇒ Z ⊂ C(G) ⇐⇒P/G where P is given in (i).

PROOF. (i) “⇒”: let P∈Syl_p(G), then P∩Z= {1} ⇒G= ZP=PZ.

(i) “⇐”: there is a unique decomposition A= Z×W with p - |^Z|and W a p-group. Then Z/G and G/Z=WP is a p-group. 

The main goal of this section is to prove

Theorem 3.52(Brauer). Any complex character of G is an integral combination of monomial characters induced from elementary subgroups of G.

The proof consists of two main steps. To state them we need Definition 3.53. The group of generalized characters is defined as

ch(G):= ^M_Zχi.

Also forF being a family of subgroups of G, we define

ch_F(G):= {Z-combinations of ψ^Gwhere ψ∈ch(H), H∈_F}. Since ψ^Gχ= (ψχH)^G, we see that ch_F(G)is an ideal of ch(G).

The first step is reduce to quasi-elementary subgroups:

Theorem 3.54. LetQ be the family of all quasi-elementary subgroups of G, then ch(G) =ch_Q(G).

Lemma 3.55. Let S6=∅ be a finite set, R⊂_Z^Sbe a “subrng”.

If R is not a subring, i.e. 1_S 6∈R, then there exists x ∈ S and a prime p such that p| f(x)for all f ∈ R.

PROOF. For x∈S, consider I_x := {f(x) | f ∈ R} ⊂Z as a subgroup.

If for all x∈ S we have Ix =_{Z, say fx}(_x) =_{1, then}

∏

Since R is a rng, expanding this out we get 1_S∈ R.  Lemma 3.56. For every g ∈ G and a prime p, there is a p-quasi-elementary subgroup H ⊂G such that p-^χ_1H^G(g).

PROOF. Writehgi =Z×W, p - |^Z|,|W| = p^k, k ≥0. Let N = N_G(Z), H ∈ Syl_p(N/Z)which containshgi/Z. That is, hgi ⊂ H ⊂ N and H = H/Z. This implies that H is a p-quasi-elementary subgroup.

From Example3.42-(2), we have

χ_1HG(g) =# Fix(π(g)), Fix(π(g)) = {aH|gaH= aH}.

All these fixed cosets lies in N: indeed a⁻¹ga ∈H ⇒a⁻¹Za⊂ H. However, since H is p-quasi-elementary, Z ⊂ H is the only subgroup with order|Z|. Hence a⁻¹Za= Z and then a∈ N.

Consider the action ofhgion N/H by left multiplications. Since Z/N, we get the induced action of W = hgi/Z on N/H. As |W| = p^k, every non-trivial orbit of it has order p^efor some e ≥ 1. (If k = 0 then there are no non-trivial orbits.) This implies

χ_1HG(g) ≡ [N : H] (mod p).

By our construction, H contains a Sylow p subgroup of N, hnece p- [^{N : P}].

This implies that p-^χ_1H^G(g). 

PROOF OFTHEOREM3.54. It is enough to show that χ_1G ∈ ch_Q(G). Let R⊂ch_Q(G)be the subrng generated by all χ_1HG with H ∈_Q.

Now comes the key point: R ⊂ _Z^|G|, instead of just C^|G|. If χ1G 6∈

ch_Q(G)then χ_1G 6∈ R. Lemma3.55then implies there exists g ∈ G and a prime p such that p|χ(g)for all χ∈ R. But this contradicts to Lemma3.56 for some χ_1H^G with H∈Q. The theorem is proved.  The second step is to reduce to elementary subgroups. We shall need:

Theorem 3.57(Blichfeldt–Brauer). Let χ be an irreducible character of p-quasi-elementary group G, then

(1) deg χ= pⁿis a p-power.

(2) χ=λ^Gfor a linear character λ, i.e. deg λ=1, of some H⊆G.

5. BRAUER’S THEOREM ON INDUCED CHARACTERS 55

PROOF. Let χ=χ_ρ, G =ZP, Z/G being cyclic and p- |^Z|.

(1) Let σ be an irreducible component of ρZ and let T = _T(σ) _{be its} inertia group. Corollary 3.49-(4) implies that ρ = τ^G for an irreducible representation τ of T. Since[G : T] = p^sand T is also p-quasi-elementary, if T6=G then (1) follows by induction on|G|_.

If T = G, notice that Z is cyclic (abelian) implies that deg σ = 1.Then Clifford’s theorem implies that a ∈ Z acts as scalar multiplication on V = V_ρ. Thus ρ is irreducible implies ρPis irreducible. Hence deg ρ | |P|_which is is a p-power.

(2) Let deg χ = pⁿ. We prove the result by induction on n. The case n=0 is trivial, so let n≥1.

For any linear character λ of G, χλ=χ⇐⇒

1= (χ, χλ) = ¹

|G|

∑

That is, the multiplicity of λ in ρ⊗ρ^∗is 1.

LetΛ := {λ|deg λ=_{1, χλ}=χ}.Λ is a group under multiplications.

Now we consider the decomposition into irreducible characters:

χχ =

∑

∑

By evaluating at g= 1, (1) implies that p| |_Λ|. Hence there exists λ₁ ∈_Λ\ {χ₁}with λ₁^p = χ₁, i.e. λ₁ : G →_C^×has its image inhζ_pi. So G/K ∼= hζ_pi for K :=ker λ. By restricting to K we get

χ_Kχ_K =

∑

∑

while λ_1K =χ_1K. So(χ_K, χ_K) = (χ_Kχ_K, χ_1K) ≥2 and then ρ_Kis reducible.

Let τ be an irreducible component of ρK. Then ρ is an irreducible com-ponent of τ^G by Frobenius reciprocity. Also deg χτ < deg χ and both are p-powers. Hence p deg χ_τ ≤ deg χ. But we have seen that[G : K] = p, so deg χ_τG = p deg χτ. This implies χ=χ_τG.

Now (2) follows by induction (on n) and transitivity of induction (on

representations). 

Lemma 3.58. Let G = ZP be a p-quasi-elementary decomposition and let W = C_G(P) ∩Z. Then H :=WP=W×P⊂ G is an elementary subgroup.

If λ : G→_C^×has λ|_H =1 then λ=χ₁.

PROOF. It is clear that P is a normal Sylow-p subgroup of H = WP, hence H=W×P is p-elementary.

For the second statement, we need only to show that λ|_Z =1.

Let K =_Z∩ker λ. Since λ is a homomorphism, we have λ(_b⁻¹_dbKd⁻¹) = 1. That is, d(bK)d⁻¹ = bK. Take b ∈ Z, d ∈ P, this implies that P acts on bK by conjugation. Since p - |^Z|, we also have p - |^K| = |bK|. As P is a p-group, this implies that the action of P on bK has a fixed point bk. Namely, bK∩C_G(P) 6= _{∅. Since}

bK∩C_G(P) ⊂Z∩C_G(P) =W ⊂ H,

we have 1=λ(bk) =λ(b). This applies to every b ∈Z, hence λ|_Z =1.  Now we can complete the second step, and hence Brauer’s theorem.

Theorem 3.59. Any character χ of a p-quasi-elementary group G is aZ-combination