We describe the linear groups and discuss the action of the projective groups on the points of the projective space. We start by defining a linear transformation on a vector space. For the classical background on symplectic forms and symplectic groups, the reader should consult [3, 68, 26, 1, 106, 118].
LetV and W be two vector spaces over a field F.A vector space homomorphism T : V −→ W such that T(v1 +v2) = T(v1) + T(v2) and T(cv) = cT(v) for all c ∈ F and v, v1, v2 ∈ V, is called a linear transformation. The set of all linear transformations T : V −→ W is denoted HomF(V,W), and if V = W, the homomorphism T : V −→ V is called an endomorphism and HomF(V,V) is denoted by EndF(V). A linear transformation T which is one-to-one and onto is called invertible.
2.6.1 The general linear group
Definition 2.17. The general linear group denoted GL(V), is the group of all invertible endomorphisms on a vector space V over a field F under addition and composition of maps.
IfV is a vector space of dimension n over a finite field F=GF(q) by choosing a basis, we can identify the vector spaceV with the spaceFnq of then-tuples of elements of F. In this case any linear transformation ofV can be represented by an invertible n ×n matrix acting on F by right multiplication. It can be seen that the general
CHAPTER 2. GROUPS 15 linear group is in fact the group of all invertible n×n matrices over F with matrix multiplication and is denoted by GL(n, q). For n= 1 we have GL(1,F) ∼=F∗ where F∗ is the multiplicative group of the field. The general linear group is generally not a simple group apart from some exceptional cases.
LetV be a n-dimensional space over F.Suppose we denote by [v] ={av:a∈F} the line through the origin spanned by a vector v ∈V. Then [v] is called a projective point. The set of all projective points in V has dimension q− 1 and is called a projective space based on V. We denote this space byP(V) or P G(n−1, q).
The elements ofGL(n, q) map a subspace to another subspace of same dimension.
ThereforeGL(n, q) acts naturally on the points of the projective space P G(n−1, q) where the action is given by T[v] = [T v] for all T ∈ GL(n, q). This action is faithful and the kernel of the action is Z(GL(n, q)). It has been shown (see [26, Proposition 2.1] that
Z(GL(n, q)) = {cIn:c6= 0, c ∈F}
where In is the n×n identity matrix, is a normal subgroup of GL(n, q) consisting of all scalar matrices of GL(n, q). This is a cyclic group whose order is q−1. If we factor out GL(n, q) by the kernel we get the projective group. So we define:
Definition 2.18. The quotient group of GL(n, q) by its center is called the projective general linear group and is denoted P GL(n, q).
Therefore
P GL(n, q) =GL(n, q)/Z(GL(n, q)) =GL(n, q)/F∗.
Theorem 2.19.
|GL(n, q)|= (qn−1)(qn−q). . .(qn−qn−1)
|P GL(n, q)|= |GL(n, q)|
(q−1)
Proof: The rows of an invertible matrix are linearly independent and will be determined by the image of its ordered basis. The (k−1)-th vector spans a vector
CHAPTER 2. GROUPS 16 space of dimension k − 1. This vector space has qk−1 vectors. The k-th vector must be independent of the previous ones and therefore will have qn−qk−1 choices.
Multiplying these for 1≤k ≤n we get the result.
The second part follows since there are q−1 non-zero scalars in F.
2.6.2 Projective special linear group
Among the important groups constructed from the general linear group are the special linear group and projective special linear group. Thespecial linear groupis the subgroup ofGL(V) consisting of all unimodular linear transformations or in terms of matrices the group of all n×n matrices with determinant 1. It is denotedSL(V) or SL(n, q). Equivalently if we define the determinant map φ : GL(n, q)−→F by φ(v) = det(v) then since
φ(uv) = det(uv) = det(u)det(v) =φ(u)φ(v)
we have that φ is an onto homomorphism. The Ker φ = {v ∈ GL(n, q) : det(v) = 1F} is a normal subgroup of GL(n, q) consisting of the matrices whose determinant is 1. This subgroup is in fact the special linear group. We thus have SL(n, q)EGL(n, q) and soGL(n, q) is generally not a simple group.
SL(n, q) induces an action on the projective space P G(n−1, q) whose kernel is the subgroup
Z(SL(n, q)) ={cIn : 06=c∈F and cn = 1}
normal in SL(n, q) consisting of all scalar matrices with determinant 1. The order of Z(SL(n, q)) is found by taking gcd(q−1, n).
Theprojective special lineargroupP SL(n, q) is defined as the quotient group of SL(n, q) by its center i.e.,
P SL(n, q) = SL(n, q)
Z(SL(n, q)) = SL(n, q)
{cIn : 06=c∈F and cn= 1}
Theorem 2.20. The groups SL(n, q) and P SL(n, q) have order:
|SL(n, q)|=|GL(n, q)|/(q−1)
|P SL(n, q)|=|SL(n, q)|/(n, q−1),
CHAPTER 2. GROUPS 17 repectively, where (n, q−1) is the gcd of n and q−1.
Proof: The proof is clear from the definition and Theorem 2.19.
In this thesis we shall use the notation of the ATLAS of finite groups [34], and thus write Ln(q) for P SL(n, q).
Theorem 2.21. The projective special linear groups P SL(n, q) is a simple group except for P SL2(2) and P SL2(3).
Proof: The groups P SL(2,2)∼= S3 of order 6 which has A3 as a normal subgroup and P SL(2,3)∼=A4 which has a normal subgroup of order 4.See [111, Theorem 4.5]
for the proof of simplicity for the remaining groups.
Theorem 2.22. The groupSL(n, q)and therefore P SL(n, q)acts double transitively on the points of the projective geometry P G(n−1, q).
Proof: Let [x] 6= [y] and [x0] 6= [y0] be two ordered pairs of points in P G(n−1, q).
(Recall that projective points are simply lines in the underlying vector spaceV(n, q)).
Then {x, y} and {x0, y0} are both linearly independent pairs in V(n, q) so we may choose them as the first two elements of the basis of V(n, q). Then we choose g ∈GL(n, q) such that g(x) = x0 and g(y) =y0. If this g ∈GL(n, q) we have chosen has det(g) =λ6= 1 then we may replace y byλy and repeat the argument to obtain
¯
g ∈GL(n, q) with det(¯g) = 1. Since [λy] = [y], it follows that ¯g ∈P SL(n, q) has the same property asg. This shows that the groupSL(n, q) and thereforeP SL(n, q) act doubly transitively on the points of the projective geometry P G(n−1, q).