Chapter 3 – Elementary Theories of Representations and Characters
Henceψ↑KK2
1↑KK3
2 =ψ↑KK3
1.The proof of the general case can be proceeded by Mathematical Induction
on n.
We conclude this section by remarking that the operations of restriction and induction of characters do not necessarily preserve irreducibility of characters.
where
φ0(y) =
1 if y∈H, 0 ify6∈H.
Hence χP =1↑GH.This shows that for any subgroupH,there exists a permutation character of G.
Conversely, ifGacts transitively on any setX,then the associated permutation character represents 1↑GH for some subgroupH ofG. This is the assertion of the following theorem.
Theorem 3.6.1. Let G act transitively on a set Ω and let ω ∈Ω. Then 1↑GG
ω is the permutation character of the action.
PROOF. Since G act transitively on Ω,we have ωG = Ω. It follows by theOrbit-Stabilizer Theorem (see Moori [49] for example) that there is a 1−1 correspondence between Ω and the set of left cosets ofGω in G,given by ωt7−→tGω fort∈G. Now forg∈G we have
(ωt)g =ωt⇐⇒ωt−1gt =ω⇐⇒t−1gt∈Gω ⇐⇒tGω=gtGω ⇐⇒tGω= (tGω)g,
where G acts on the set of left cosets of Gω in G as given above. Therefore the permutation character of the action of G on Ω is the same as the permutation character of the action of G on the left cosets of Gω inG, which is 1↑GG
ω.
Corollary 3.6.2. Let G act on Ω with a permutation character χ. Suppose Ω decomposes into exactlyk orbits under the action of G. Then hχ,1i=k, where 1 is the trivial character of G.
PROOF. Write Ω =
k
[
i=1
∆i where the ∆i are orbits. Letχi be the permutation character ofG on ∆i
so that χ=
k
X
i=1
χi.For ω∈∆i,we have χi =1↑GG
ω by Theorem 3.6.1. Thus hχi,1iG=
1↑GG
ω,1
G= 1,1↓GG
ω
Gω = 1 by Frobenius reciprocity. Therefore hχ,1i=
k
X
i=1
hχi,1i=k,completing the proof.
Lemma 3.6.3. If Gacts transitively onΩ,then all subgroupsGω, ω∈Ωof Gare conjugate in G.
PROOF. SinceGacts transitively on Ω,there is someh∈Gsuch thatωh =κfor anyω, κ∈Ω.Now g∈Gω ⇐⇒ωg =ω ⇐⇒κgh−1 =κh−1 ⇐⇒κhgh−1 =κ⇐⇒hgh−1 ∈Gκ⇐⇒g∈(Gκ)h.
Chapter 3 – Elementary Theories of Representations and Characters
ThusGω = (Gκ)h,which shows thatGω =hGκh−1.That isGω and Gκ are conjugate inG.
Because 1↑GH is a transitive permutation character, it must satisfy certain necessary conditions mentioned in the following theorem.
Theorem 3.6.4. Let H≤G and χ=1↑GH.Then (i) deg(χ)||G|.
(ii) hχ, ψi ≤deg(ψ), ∀ψ∈Irr(G).
(iii) hχ,1i= 1.
(iv) χ(g)∈N∪ {0}, ∀g∈G.
(v) χ(g)≤χ(gm), ∀g∈G, ∀m∈N∪ {0}.
(vi) o(g)-χ(1|G|G) =⇒χ(g) = 0.
(vii) χ(g)χ(1|[g]|
G) ∈Z, ∀g∈G.
PROOF. Let Ω be the set of the left cosets of H inG. Thus χ is the permutation character ofG on Ω.
(i) Since deg(χ) = [G:H],we have deg(χ)||G|.
(ii) Using Frobenius reciprocity we gethχ, ψiG=
1↑GH, ψ
G=
1↓GH, ψ↓GH
H ≤deg(ψ).
(iii) Since χis a transitive permutation character, it follows by Corollary 3.6.2 that hχ,1i= 1.
(iv) This follows because χ(g) is the number of points left fixed byg and hence is non-negative.
(v) Let g∈Gω, that is ωg =ω. It is clear thatωgm =ω. Thus any point of Ω left fixed by g is also fixed by gm. Therefore the number of points fixed by g does not exceed the number of points fixed bygm.
(vi) We know that χ(1|G|
G) =|H|so ifo(g)-|H|,then [g]∩H =∅,the empty set. Hence1↑GH(g) = 0.
(vii) Let B={(ω, x)|ω∈Ω, x∈[g], ωx =ω}.Since χis constant on [g],we have
|[g]|χ(g) =|B|= X
ω∈Ω
|[g]∩Gω|.
By Lemma 3.6.3 all subgroupsGω are conjugate in G.Thus|[g]∩Gω|=mis independent of ω, and χ(g)|[g]|=m|Ω|=mχ(1G).
This completes the proof.
Corollary 3.6.5. Let H ≤ G with χ = 1↑GH. Let g ∈ G and assume that [g] splits in H into m classes with representatives h1, h2,· · · , hm.Then
χ↑GH(g) =
m
X
i=1
|CG(g)|
|CH(hi)|.
PROOF. Immediate by Proposition 3.5.5.
Chapter 4
Schur Multiplier, Projective Representations and Characters
In this chapter we introduce the concept of the projective representations and characters of a finite group G, which are of great importance in the sequel of this thesis. We also introduce the Schur multiplier M(G) of a groupG, which plays an important role in the study of projective representations and characters ofG.In Section 4.1, we give some definitions and basic results on the Schur multiplier ofG.Section 4.2 is devoted to the theory of projective representations ofG.There is a relationship between the projective representations ofGand the ordinary representations of the central extension M(G)·G, which is known ascovering group orrepresentation group of G. Every projective representation can be obtained from (or we say lifted to) an ordinary representation of M(G)·G. In fact if α is factor set of the Shcur multiplier ofGand ξ is a projective representation of G with associated factor set α, then ξ can be lifted to an ordinary representation of α·G and we do not need to calculate the full covering group. In Section 4.3 we discuss the projective characters and study the orthogonality relations analogous to the ones for ordinary characters, given by Theorem 3.2.3. For further readings on projective representations and projective characters readers are referred to Berkovich and Zhmud [15], Haggarty [32], Hoffman and Humphreys [34], Humphreys [35], Huppert [37], Isaacs [38], Karpilovsky [42], [43], Morris [51], [52], [53], [54], Nagao and Tsushima [56], Read [59], [60] and [61].
4.1. Schur Multiplier
The Schur multiplier of a finite group G is of particular interest and importance to study all the projective representations of G. In this section we discuss some results that are useful in finding the Schur multiplier ofG.
Definition 4.1.1. A function α:G×G−→C∗ is called a factor set ofG if
α(xy, z)α(x, y) =α(x, yz)α(y, z), ∀ x, y, z∈G. (4.1) Example 4.1.1. A trivial example of a factor set of any finite group G is the map 1, where 1(x, y) = 1, ∀x, y∈G.
Next we define an equivalence relation amongst factor sets of a group G. Two factor sets α and β are said to beequivalent if there exists a functionρ:G−→C∗ such that
β(x, y) = ρ(x)ρ(y)
ρ(xy) α(x, y), ∀ x, y∈G. (4.2) Clearly the above relation defines an equivalence relation amongst the factor sets. We denote the equivalence class containing the factor set α by [α].We define multiplication of two factor sets α and β by (αβ)(x, y) = α(x, y)β(x, y), ∀ x, y ∈ G. With no much difficulty we can see that αβ satisfies Eq. (4.1) and thusαβ is again a factor set. Moreover, we can see that for every factor set α,the functionα−1:G×G−→C∗,defined byα−1(x, y) = (α(x, y))−1= α(x,y)1 is also a factor set of G.
Definition 4.1.2. The set of all equivalence classes of factor sets of a group G forms a group by defining [α][β] = [αβ].The identity of this group is [1] and [α]−1 = [α−1].This group is called the Schur multiplier or multiplicator of G and we denote it by M(G).
Next we quote some results from Karpilovsky [42] on the Schur multipliers of finite groups.
Theorem 4.1.1. (i) M(G) is a finite abelian group.
(ii) If G is a cyclic group, then M(G) = 1.
PROOF. See Nagao and Tsushima [56] or Rotman [64].
Lemma 4.1.2. Suppose that N is a normal subgroup of a finite group G. If M(G) = 1, then M(G/N)∼= (NT
G0)/[N, G].In general, |(NT
G0)/[N, G]| divides|M(G/N)|.
Chapter 4 – Schur Multiplier, Projective Representations and Characters
PROOF. See Karpilovsky [43].
Theorem 4.1.3. LetGbe a finite group andH be a subgroup of indexm.Then(M(G))m ∼=M(H), where (M(G))m is the group of allmth powers of M(G).
PROOF. See Karpilovsky [43].
Schur [68] reduced the problem of computingM(G) to compute the Schur multiplier of the Sylow p−subgroups ofG,wherepis a prime dividing the order ofG.The following theorem describes the Schur multiplier of a groupG in terms of the Sylowp−subgroups ofG.
Theorem 4.1.4 (Schur [68]). Let S be a Sylow p−subgroup of G. Then the Sylow p-subgroup of M(G) is isomorphic to a subgroup of M(S).
PROOF. See Karpilovsky [43].
Theorem 4.1.5. A group G has trivial Schur multiplier if and only if it has a set of subgroups with trivial Schur multipliers and relatively prime indices.
PROOF. See Karpilovsky [43].
In GAP [30] and Magma [16] there are various commands to calculate the Schur multiplier of a finite groupG.For example the commands “AbelianInvariantsMultiplier(G)” and “pMultiplicator(G,p)”
of GAP and Magma respectively reveals the Schur multiplier of G and of a Sylowp−subgroup of Grespectively. We deal with some examples in Chapters 7 and 8.