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).
CHAPTER 2. GROUPS 18 Definition 2.23. Let V be a finite dimensional vector space over a field F. A function h,ifrom the set V ×V of ordered pairs in V toF is called a bilinear form on V if for each v ∈ V, the functions hv, i and h , vi are linear functionals on V. In this case we say that (V,h,i) is an inner product space.
Ifh,iis a bilinear form onV such that for each non-zerox∈V,there existsy∈V for which hx, yi 6= 0, then h,i is said to be non-degenerate.
Remark 2.24. A bilinear form h,i is called alternating (symplectic) on V if hx, xi= 0 for all x∈V.
Let V be a vector space over a field F and h,i be a symplectic form on V. If char(F)6= 2 then we obtain that for all x, y ∈V,
0 = hx+y, x+yi=hx+y, xi+hx+y, yi=hx, xi+hy, xi+hx, yi+hy, yi.
However, since hx+y, x+yi = hx, xi = hy, yi = 0 we have that hx, yi = −hy, xi.
Conversely ifh,i is a bilinear form for whichhx, yi=−hy, xifor allx, y ∈V, then in particular for x ∈V we have hx, xi =−hx, xi. This implies that 2hx, xi= 0 and so hx, xi= 0, for all x∈V.
Definition 2.25. Let V be a vector space over a field F. Let h,i:V ×V −→F be a bilinear form on V such that
(i) hx, xi= 0, for all x∈V
(ii) hx, yi=−hy, xi, for all x, y ∈V.
Then we say that (V,h,i) is a symplectic space over the field F.
Remark 2.26. Ifchar(F)6= 2, then the properties (i) and (ii) in the above definition are equivalent.
Let (V,h,i) and (U,h,i) be symplectic spaces over F, then we say that V ∼=U if there exists an isomorphism T ∈ HomF(V, U) such that for all x, y ∈ V we have hx, yi=hT(x), T(y)i.
Definition 2.27. Let (V,h,i) be a symplectic space. If x, y ∈ V, then x and y are orthogonal if hx, yi = 0. If W is a subspace of V then the orthogonal
CHAPTER 2. GROUPS 19 complement of W is defined by
W⊥={y∈V | hx, yi= 0, f or all x∈W}.
Note 2.28. Note that for all x ∈W we have h0, xi=hx−x, xi=hx, xi − hx, xi= 0−0 = 0, so that 0 ∈W⊥. Now if x, y ∈W⊥, then for any α, β ∈ F and z ∈ W we have
hαx+βy, zi=hαx, zi+hβy, zi=αhx, zi+βhy, zi=α0 +β0 = 0, therefore αx+βy ∈W⊥, and henceW⊥ is a subspace ofV.
Let (V,h,i) be a symplectic space and define R(V) by R(V) = V⊥. Then R(V) is called the radical of V. We can easily see that (V,h,i) is non-degenerate if and only if R(V) = {0V}.
Definition 2.29. In an inner product space (V,h,i), a vector v is called isotropic if hv , vi = 0. A subspace U of V is called isotropic if there exists in U a non-zero vector z such that it is orthogonal to V.
It is clear that in a symplectic space every vector is isotropic.
Note 2.30. Definition 2.29 suggests thatU being isotropic is equivalent toh,ibeing degenerate when restricted to U. This in turn equivalent is to U ∩U⊥ 6= V. A subspace U is called totally isotropic if all its vectors are isotropic. Having U being totally isotropic is equivalent to having any two vectors in U orthogonal. This then implies that U ⊆U⊥.
Definition 2.31. Consider(V,h,i)withh,ibilinear . If{v1, v2, . . . , vm}is an ordered basis of V, then the inner product matrix of h,i relative to this basis is given by an m×m matrix A= [hvi, vji]m×m .
If (V,h,i) is a symplectic space of dimension 2m then the form is given by
hu, vi=uM vT (2.1)
CHAPTER 2. GROUPS 20 where a basis {e1, e2, . . . , e2m} for V can be chosen such that M = [mi,j] is the 2m×2m matrix given as follows: if B is the m×m matrix [bi,j] where bi,j = 1 if i+j =m+ 1, and bi,j = 0 otherwise, that is
B =
1 1 . . 1 1
,
then
M =
0 −B
B 0
. Thus
hei, eji=eiM eTj = 0 if i+j 6= 2m+ 1, and
hei, e2m+1−ii=eiM eT2m+1−i =−1 for 1≤i≤m, and
hei, e2m+1−ii=eiM eT2m+1−i = 1 for m+ 1 ≤i≤2m.
The symplectic group Sp2m(q) is the subgroup of GL2m(q) of transformations g for which hug, vgi = hu, vi, for all u, v ∈ V. That is, it is matrix Q for which QM QT = M. The projective symplectic group P Sp2m(q) is the factor group Sp2m(q)/Z(Sp2m(q)) and we have
Z(Sp2m(q)) = {I} if char(Fq) = 2 and Z(Sp2m(q)) = {I,−I} if char(Fq)6= 2.
The projective symplectic groups are simple except for P Sp2(2) = P SL2(2),
CHAPTER 2. GROUPS 21 P Sp2(3) =P SL2(3) andP Sp4(2). The order of P Sp2m(q) is given by
|P Sp2m(q)| = 1
(2, q−1)× |Sp2m(q)|
= qn2 (2, q−1)
n
Y
i=1
(q2i−1).
Now if we let P(V) denote the projective space defined by V, that is P(V) = P G2m−1(q) (see Section 4.4 for more details on projective geometry) then the symplectic form on V defines a polarity on P(V), a correspondence between the elements of P(V) that reverses inclusion and has order 2. If we denote this polarity by σ, then for any U ∈ P(V) we have σ:U 7→Uσ where
Uσ ={v | v ∈V, uM vT = 0, f or all, u∈U}. (2.2) It can be shown thatP Sp2m(q) is the group of all the collineations ofP(V) that commute with the polarity σ.
In the context of projective space, the subspace U ∈ P(V) is called totally isotropicifU∩Uσ =U,isotropicifU∩Uσ 6=∅andnon-isotropicifU∩Uσ =∅. We can see that for symplectic polarity, points are always totally isotropic. Any totally isotropic space has dimension at most m, and those subspaces of dimension m are called maximal isotropic subspaces. A point P of P(V) is said to be absolute if P lies on Pσ.
IfP is a point of the projective (2m−1)-spaceP G2m−1(q) then the affine subgroup of G=P Sp2m(q) is the stabilizer GP of the form N :P Sp2m−2(q), a split extension, where N is a p-group of orderq2m−1 (see [57]).
Ifq=prwherepis an odd prime,N will be a non-abelian specialp-group of order q2m−1. If p = 2, then N is an elementary abelian 2-group. For further information on the affine subgroups of the symplectic group, see ([101, Chapter 10]).
The simple symplectic groupP Sp2m(q), where mis at least 2 and q is any prime power, acts as a primitive rank-3 group of degree q2mq−1−1 on the points of the projective (2m−1)-space P G2m−1(Fq), (see Theorem 2.32). The orbits of the stabilizer of a point P consist of{P}and one of length q2m−1q−1−1 −1 and the other of length q2m−1.
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].