Selección de Modelos de Capacidad

We will require an understanding of linear and classical groups in Section 3.4 and Chapter 4. In this section, we remind the reader of how these groups are defined and put in place the notation that will be in force for the rest of the thesis; the reader is referred to [41, Chapter 3] for the proofs of any standard facts.

Suppose that V is an n-dimensional vector space over a finite field F =_Fq of order q for some prime power q= pr with _{r ∈}

N. The general linear group, denoted by GLn(q), consists of all the invertible linear transformations φ:V →V. We denote by SLn(q) the special linear group, which consists of those elements in GLn(q) with

determinant 1, and by Ln(q) the projective linear group, which is defined to be the

quotientLn(q) :=SLn(q)/Z(SLn(q)). The general linear groups, special linear groups

and projective linear groups are sometimes referred to loosely aslinear groups. The other classical groups arise as subgroups of general linear groups which fix certain extra structures on the relevant vector space. Recall that a bilinear form is a map f :V ×V →F such that f(λu+v, w) =λf(u, w) +f(v, w) andf(u, λv+w) = λf(u, v) +f(u, w) for all λ ∈ F and u, v, w ∈ V. We say that f is symmetric if

f(u, v) = f(v, u) for all u, v ∈ V; skew-symmetric if f(u, v) = −f(v, u) for all u, v ∈V; and alternating if f(v, v) = 0 for all v ∈V. If F = _Fq2, then we setx= xq

1.2 Group Theory 25 for all x∈F and we shall say that a map f :V ×V →F satisfying

f(λu+v, w) =λf(u, w) +f(v, w)

and f(u, v) =f(v, u)

for all u, v, w ∈V and λ ∈F is a conjugate-symmetric sesquilinear form.

Given a bilinear form or a sesquilinear form f defined on V, we set Isom(V, f)

to denote the set of φ ∈ GLn(q) such that f(φ(u), φ(v)) = f(u, v) for all u, v ∈ V

and call this the isometry group of V with respect to f. Furthermore, we set

Sim(V, f) to be equal to the set of φ∈GLn(q) for which there exists λ∈F such that f(φ(u), φ(v)) = λf(u, v) for all u, v ∈ V; this is the similarity group of V with respect to f.

We are now ready to define the three main types of classical groups; in the following, we denote the (n×n)-identity matrix byIn. If dimV = 2m, then Sp2m(q) denotes the symplectic group of degree 2m over F, which is the isometry group of a nonzero

alternating bilinear form f on V. It is well-known that, up to isomorphism, there is

one such group in any even dimension and for any finite field F. We defineP Sp2m(q)

to be the quotient Sp2m(q)/{±I2m}and call this the projective symplectic group;

we also setGSp2m(q) to equal the similarity group of V with respect tof.

If dimV = n and F =_Fq2, then GU_n(q) denotes the general unitary group of

degree n over F, which is the isometry group of a conjugate-symmetric sesquilinear

formf onV; note thatGUn(q)≤GLn(q2). We denote bySUn(q) the special unitary group, i.e.,SUn(q) =GUn(q)∩SLn(q2); and byUn(q) theprojective unitary group,

that is to say, the quotient SUn(q)/Z(SUn(q)).

If dimV = n is odd and F = _Fq with charF > 2, then On(q) denotes the orthogonal group of degree n over F, which is the isometry group of a nonzero

symmetric bilinear form f on V. In this case, there is up to isomorphism just one

orthogonal group. On the other hand, if dimV = 2m is even, then we get two different

orthogonal groups, depending on certain special properties of the bilinear form f. We

say that f is of plus type if there exists a subspaceU in V of dimension m such that f is identically zero onU ×U, and it is of minus typeotherwise. The orthogonal groups O+_2m(q) and O₂−_m(q) are then the isometry groups corresponding to nonzero

26 Preliminaries symmetric bilinear forms of the obvious type. When we wish to discuss all three of the above cases generically, we shall revert to the symbol On(q), with the understanding

that for even n, this notation is potentially ambiguous.

Forϵ∈ {+,−,∅}, we denote bySOϵ

n(q) thespecial orthogonal groupcorrespond-

ing toO_nϵ(q), i.e.,SOϵ_n(q) =O_nϵ(q)∩SLn(q); by Ωϵn(q), we denote the kernel of the spinor

norm map, which is an index 2-subgroup ofSOϵ

n(q); and we setPΩϵn(q) := Ωϵn(q)/{±In}

when n is even, and PΩϵ_{(q) := Ω}ϵ

n(q) if n is odd. For details concerning the spinor

norm map, the reader should refer to [41, 3.7].

Finally, we need to define the orthogonal groups over fields of characteristic 2. If dimV =n is even and greater than or equal to 6, and F =_Fq with charF = 2, then

for each vector v ∈V of norm 1, the map defined by tv :w7→w+f(w, v)v

in terms of a nonzero symmetric bilinear form f on V is known as an orthogonal transvection. The orthogonal group O+

n(q) is defined to be the subgroup of GLn(q) generated by these transvections. Moreover, thequasideterminantof a given x∈O+

n(q) is defined to be 1 or−1 according to whetherx can be written as a product

of either an even number or an odd number of these transvections. The set of elements which are of quasideterminant 1 then form a subgroup of O+

n(q) which we denote

either Ω+

n(q) or, for the purposes of being consistent with our terminology for the

simple classical groups,PΩ+

n(q). A similar construction provides another corresponding

orthogonal group of “minus” type, which we denote by O₂−_m(q), and this too contains

a subgroup generated by certain transvections, which we denote by either Ω−

n(q) or PΩ−_n(q). We refer the reader to the discussion contained in [41, 3.8] for further details

on the constructions for bothO+

n(q) and O

−

n(q) in the case where F has characteristic

2.

We refer to any of the above symplectic, unitary or orthogonal groups as aclassical group. Of course, the classical groups play an important role in the classification of the finite simple groups; the following result clarifies which of the above groups are simple.

1.2 Group Theory 27 Theorem 1.2.3. The following classical groups are all simple:

(i) Ln(q) if n >2 or q >3;

(ii) P Sp2m(q) if m= 1 and q >3, or m= 2 andq >2, or m >2;

(iii) Un(q) if n= 2 andq >3, or n = 3 andq >2, or n >3;

(iv) forϵ∈ {+,−,∅}, PΩϵ

n(q) if n= 5 and q is odd, or n≥6.

In addition to the above theorem, we have the following well-known isomorphisms between classical groups of low degree:

L2(2)∼= Σ3 L2(3) ∼=A4 L2(4) ∼=L2(5)∼=A5 L2(7) ∼=L3(2) L2(9)∼=A6 L4(2)∼=A8.

At times, we shall provide examples which involve the classical groups described above, and when we do so, we shall find it helpful to think of them as being groups of matrices. Thus, we may think ofGLn(q) as consisting of the (n×n)-matrices with entries

in a finite field of orderq, and we denote this field by _Fqand its multiplicative subgroup

by F×q. By a diagonal matrix, we mean a matrix with diagonal (α1, α2, . . . , αn) for

some αi ∈F×q, and zeroes everywhere else; and by ascalar matrix, we mean a matrix

with diagonal (α, α, . . . , α) for some fixed α ∈ _F×

q, and zeroes everywhere else. We

denote by eij the matrix with a 1 in its (i, j)-th entry, and zeroes everywhere else. A permutation matrix is a matrix which has precisely one entry equal to 1 in every row and column, and zeroes everywhere else; a monomial matrix is a matrix which is a product of a diagonal matrix and a permutation matrix.