CHAPTER 2. GROUPS 22 Theorem 2.32. If m ≥2, then P Sp2m(q) acts as a primitive permutation group of rank-3 on the points of P(V).
Proof: See [111, Theorem 8.2 and Theorem 8.3].
CHAPTER 2. GROUPS 23
Now
(An){σ
1, σ2} = {g ∈An | {σ1, σ2}g ={σ1, σ2}}
= {g ∈An |σ1g =σ1, σ2g =σ2 or σ1g =σ2, σ2g =σ1}.
Clearly
(An)[σ1,σ2]=An−2 ≤(An){σ
1, σ2}
and
K ={(σ1 σ2)·α|α ∈(Sn)[σ1,σ2], α is odd} ⊆(An){σ
1, σ2}. Thus An−2∪K ≤(An){σ
1, σ2} and
|An−2∪K|=|An−2|+|K|= 2|An−2|= 2(n−2)!
2 = (n−2)! =|(An){σ
1, σ2}|, by 2.3 . Hence (An){σ
1, σ2} =An−2∪K. Since (An){σ1,σ2} ≤An, |(An){σ
1, σ2}|= 2|An−2|=|Sn−2| and An−2 = (An)[σ1,σ2]≤(An){σ1,σ2},we can deduce that (An){σ
1, σ2} ∼=Sn−2. The group (An){σ
1, σ2} has three orbits {{σ1, σ2}}, {σi, γ|i ∈ {1,2}, γ ∈ Ω\ {σ1, σ2}} and {γ, µ|γ, µ ∈ Ω\ {σ1, σ2}, γ 6= µ}. These orbits have lengths 1, 2(n −2) and (n−2)(n−3)2 , respectively. Now any non-trivial block for the action of An on Ω{2} which contains the point {σ1, σ2} must also contain one of the other orbits of (An){σ
1, σ2}. However, a simple argument shows that for n6= 4 such a block must also contain the other orbit, and so the action of An on Ω{2} is primitive. Now since (An){σ
1, σ2} is the stabilizer of a point in the action of An on Ω{2} and An is primitive we have that (An){σ
1, σ2} is maximal.
Remark 2.35. If n = 4, then Theorem 2.34 is not true since (A4){σ
1, σ2} ∼= S2 = {1S2,(σ1 σ2)(σ3 σ4)} and A4 is clearly not a rank-3 group on Ω{2} where Ω = {1,2,3,4}.
Groups that act two-transitively yield two designs as we shall see in Section 5.4.
A group G acts sharply transitively on a set Ω if its action is regular, that is, it is transitive and the stabilizer of a point is the identity.
Chapter 3
Representations and modules
In this section we give some preliminary results on representations and characters of groups which will be needed in later chapters. We note that in this section F is a field and V is a finite-dimensional vector space over F. References for this section include [9, 90, 94, 72, 100].
3.1 Representations
Definition 3.1. LetGbe a finite group and let V be a vector space of dimensionn over the field F. Then a homomorphismρ:G −→ GL(n,F) is said to be a matrix representation of G of degree n over the field F. The column space, Fn×1 of ρ is called the representation module of ρ. If the characteristic of F is zero then ρ is called an ordinary representation while a representation over a field of non-zero characteristic is called a modular representation.
Remark 3.2. (i) A representation ρ : G −→ GL(n,F) is said to be injective if the kernel Ker(ρ) = {1G}. An injective representation is called a faithful representation in which case G ∼= Im(ρ) so that G is isomorphic to a subgroup of GL(n,F).
(ii) Every group has a degree 1 matrix representation ρ:G −→ GL(1,F) = F∗ defined by ρ(g) = 1F for all, g ∈ G. This representation is called the trivial
24
CHAPTER 3. REPRESENTATIONS AND MODULES 25
representation.
Definition 3.3. Let ρ : G−→ GL(n,F) be a representation of G over the field F. The function χ : G −→ F defined by χ(g) = trace(ρ(g)) is called the character of ρ. If φ : G → F is a function that is constant on conjugacy classes of G i.e., φ(g) =φ(αgα−1) for all,α∈Gwe say that φis aclass function. It is easily shown that any character χ is a class function.
Recall from linear algebra that, if V is a finite dimensional F-vector space then GL(V) ∼= GL(n,F). Hence given any g ∈ G and a representation %:G −→
GL(V), %(g) ∈ GL(V) and if we let B = {v1, . . . , vn} be a basis for V then we obtain that the corresponding matrix representation ρ(g) ∈ GL(n,F) with respect to the basis B is given by ρ(g) = [aij] where
ρ(g)(vj) =
n
X
i=1
aijvi.
Similarly, if we are given an invertible matrix representation ρ:G−→ GL(n,F) then for ρ(g) ∈ GL(n,F) it follows that we can define a representation %:G −→
GL(V) by%(g)(v) = ρ(g)v wherev ∈Fn×1 is a column vector in the column space of ρ(g) with respect to the standard basis. Seeing that we can describe a representation in terms of a matrix with respect to some basis, it is clear that we should be able to define what it means for two representations to be equivalent.
Definition 3.4. Two matrix representations ρ1 and ρ2 of G are said to be equivalent representations if there exists P ∈ GL(n,F) such that ρ2(g) = P ρ1(g)P−1 for all, g ∈G.
Thus, whenever we consider a representation, it is only considered up to equivalence. We next discus the concepts of reducibility and decomposability of representations.
Definition 3.5. Let ρ :G−→GL(n,F) be a representation of G on a vector space V =Fn.LetW ⊆V be a subspace of V of dimensionmsuch thatρg(W)⊆W for all, g ∈G, then the map G→GL(m,F) given by g 7−→ρ(g)|W is a representation of G
CHAPTER 3. REPRESENTATIONS AND MODULES 26 called asubrepresentationof ρ. The subspaceW is then said to beG-invariant or a G-subspace. Every representation has {0} and V as G-invariant subspaces. These two subspaces are called trivial or improper subspaces.
Definition 3.6. A representation ρ:G −→ GL(n,F) of G with representation module V is called reducible if there exists a proper non-zero G-subspace U of V and it is said to be irreducible if the onlyG-subspaces of V are the trivial ones.
Suppose ρ:G −→ GL(n,F) is a reducible matrix representation with U ⊆Fn×1 and dimFU =m then we can choose a basisC for U that can be extended to a basis Bof V =Fn×1 so that ρ has the form
ρ(g) =
β(g) γ(g) 0 δ(g)
for all g ∈ G, where β(g) = ρ1(g), and δ(g) = ρ2(g) are matrix representations ρ1:G−→GL(m,F) andρ2:G−→GL(n−m,F) of Grespectively. That is the first m columns ofρ are formed by the basis C of the subspaceU.
The representation module V of an irreducible representation is called simple and the ρ-invariant subspaces of a representation module V are calledsubmodules of V. A simple subspace U of V is a submodule that is isomorphic to a simple representation module and it is called a composition factorof V.
Definition 3.7. Let ρ1 : G → GL(n,F) and ρ2 : G → GL(n0,F) be two representations, their direct sum ρ1⊕ρ2 : G→GL(n+n0,F) is defined by
g 7→
ρ1(g) 0 0 ρ2(g)
.
Definition 3.8. Let ρ :G →GL(V) be a representation of G on a vector space V.
If there exists G-invariant subspacesU andW such that V =U⊕W then ρ is called decomposable. If no such subspaces exist it is called indecomposable.
Suppose thatρ : G−→GL(n,F) is decomposable with G-invariant subspacesU and W i.e., V = U ⊕W. Let B1 and B2 be the basis of the G-subspaces U and W
CHAPTER 3. REPRESENTATIONS AND MODULES 27 respectively, then with respect to the basis B = B1 ∪ B2 the matrix representation will be of form:
[ρg]B =
[ρU(g)]B1 0 0 [ρW(g)]B1
.
Definition 3.9. A representation ρ is said to be completely reducible or semi- simple if it is the direct sum of irreducible representations.
From the definition we deduce that a completely reducible representation is reducible, however the converse is not necessarily true.