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Classical Groups Chapter 2 Classical groups 2.1 Introduction In this chapter we describe the six families of so-called ‘classical’ simple groups. These are the linear, unitary and symplectic groups, and the three families of orthog- ¢¡ ¥¤ onal groups. All may be obtained from suitable matrix groups G by taking G Z £ G . Our main aims again are to define these groups, prove they are simple, and describe their automorphisms, subgroups and covering groups. We begin with some basic facts about finite fields, before constructing the general ¤ linear groups and their subquotients PSLn £ q , which are usually simple. We prove simplicity using Iwasawa’s Lemma, and construct some important subgroups using the geometry of the underlying vector space. This leads into a description of pro- jective spaces, especially projective lines and projective planes, setting the scene for ¤ proofs of several remarkable isomorphisms between certain groups PSLn £ q and other ¤§¦¨ ¤§¦¨ ¤©¦¨ £ £ groups, in particular PSL2 £ 4 PSL2 5 A5 and PSL2 9 A6. Covering groups and automorphism groups are briefly mentioned. The other families of classical groups are defined in terms of certain ‘forms’ on the vector space, so we collect salient facts about these forms in Section 2.4, before introducing the symplectic groups in Section 2.5. These are in many ways the easiest of the classical groups to understand. We calculate their orders, prove simplicity, discuss subgroups, covering groups and automorphisms, and prove the generic isomorphism ¤ ¦ ¤ ¤ ¦ ¨ ¨ £ £ Sp2 £ q SL2 q and the exceptional isomorphism Sp4 2 S6. Next we treat the unitary groups in much the same way. Most of the difficulties in studying the classical groups occur in the orthogonal groups. The first problem is that there are fundamental differences between the case when the underlying field has characteristic 2, and the other cases. The second problem is that even when we take the subgroup of matrices with determinant 1, and factor out any scalars, the resulting group is not usually simple. To avoid introducing too many complications at once, we first treat the case of orthogonal groups over fields of odd order, defining the three different types and cal- culating their orders. We use the spinor norm to obtain the (usually) simple groups, and describe some of their subgroup structure. Then we briefly describe the differ- ences that occur when the field has characteristic 2, and use the quasideterminant to obtain the (usually) simple group. 19 20 CHAPTER 2. CLASSICAL GROUPS The generic isomorphisms of the orthogonal groups all derive from the Klein cor- ¤ ¦ ¤ ¤ ¦ ¤ ¨ Ω ¨ Ω £ £ ¡ £ £ respondence, by which P q PSL4 q (and also P q PSU4 q ). This corre- ¦ 6 6 ¤ ¨ spondence also leads to an easy proof of the exceptional isomorphisms PSL4 £ 2 A8 ¤ ¦ ¤ ¨ £ and PSL3 £ 2 PSL2 7 . We complete the chapter by proving a simple version of Aschbacher’s theorem which gives a basic classification of the different types of maximal subgroups in a classical group. This proof relies heavily on representation theory, however, and can be omitted by those readers without the necessary prerequisites. 2.2 Finite fields All the classical groups are defined in terms of groups of matrices over fields, so before we can define the finite classical groups we need to know what the finite fields are. A field is a set F with operations of addition, subtraction, multiplication and division sat- ¤ ¢¤£¥¢§¦¨¢ isfying the usual rules. That is, F has an element 0 such that £ F 0 is an abelian ¡ ¤ £ © ¢¢ ¢ group, and F © 0 contains an element 1 such that F 0 1 is an abelian group, ¨ ¤ £ £ and x £ y z xy xz. It is an easy exercise to show that the subfield F0 generated by the element 1 in a finite field F is isomorphic to the integers modulo p, for some p, and therefore p is prime (called the characteristic of the field). Moreover, F is a vector space over F0, as the vector space axioms are special cases of the field axioms. As every finite dimensional vector space has a basis of n vectors, v1, . , vn, say, and ∑n every vector has a unique expression a v with a F , it follows that the field F i 1 i i i 0 has pn elements. Conversely, for every prime p and every positive integer n there is a field of order pn, which is unique up to isomorphism. [See below.] The most important fact about finite fields which we need is that the set of all non-zero elements forms a multiplicative group, which is cyclic. For the polynomial ring F x over any field F is a Euclidean domain and therefore a unique factorization domain. In particular a polynomial of degree n has at most n roots. If the multiplicative e ¦ ¦ group F of a field of order q has exponent e strictly less than q 1, then x 1 has ¦ q ¦ 1 roots, which is a contradiction. Therefore the exponent of F is q 1, so F contains elements of order q ¦ 1, since it is abelian, and therefore it is cyclic. q ¨ q Note also that all elements x of F satisfy x x, and so the polynomial x ¦ x ¨ ¤ n ¦ α factorizes as ∏ £ x . Moreover, the number of solutions to x 1 in F is the α F ¤ ¢ ¦ ¦ greatest common divisor £ n q 1 of n and q 1. A useful consequence of this for 2 £ λ 2 the field of order q is that for every µ q there are exactly q 1 elements q ¨ λλ ¨ λ1 q satisfying µ. ¦ ¡ ¨ Observe that if f is an irreducible polynomial of degree d over the field p p ¤ ¡ d £ of order p, then p x f is a field of order p . Conversely, if F is a field of order d p , let x be a generator for F , so that the minimum polynomial for x over p is an ¦ ¡ ¤ ¨ £ irreducible polynomial f of degree d, and F p x f . One way to see that such ¨ a field exists is to observe that any field of order q pd is a splitting field for the q polynomial x ¦ x. Splitting fields always exist, by adjoining roots one at a time until q the polynomial factorises into linear factors. But then the set of roots of x ¦ x is closed ¨ ¨ ¨ ¨ q q ¤ q q q under addition and multiplication, since if x x and y y then £ xy x y xy and ¨ ¨ ¤ q q q £ £ £ £ x y x y x y. Hence this set of roots is a subfield of order q, as required. 2.3. THE GENERAL LINEAR GROUPS 21 ¨ To show that the field of order q pd does not depend on the particular irreducible ¨ ¡ ¤ £ polynomial we choose, suppose that f1 and f2 are two such, and Fi p x fi . Since ¤ q q ¦ ¦ f2 £ t divides t t, and t t factorizes into linear factors over F1, it follows that ¨ ¤ ¡ F1 contains an element y with f2 £ y 0. Hence the map x y extends to a field homomorphism from F2 to a subfield of F1. Moreover, the kernel is trivial, since fields have no quotient fields, so this map is a field isomorphism, since the fields are finite. ¨ ¨ ¨ If also f1 f2 then any automorphism of F F1 F2 has this form, so is defined by the image of x, which must be one of the d roots of f1. Hence the group of auto- ¡ p morphisms of F has order d. On the other hand, the map y y (for all y F) is an ¤ automorphism of F, and has order d. Hence Aut £ F is cyclic of order d. 2.3 The general linear groups Let V be a vector space of dimension n over the finite field q of order q. The general ¤ linear group GL £ V is the set of invertible linear maps from V to itself. Without much n loss of generality, we may take V as the vector space q of n-tuples of elements of q , ¤ ¤ £ ¢ and identify GL £ V with the group (denoted GLn q ) of invertible n n matrices over q . ¨ ¤ There are certain obvious normal subgroups of G GLn £ q . For example, the ¨ λ £ λ centre, Z say, consists of all the scalar matrices In, where 0 q and In is the ¦ n ¢ n identity matrix. Thus Z is a cyclic normal subgroup of order q 1. The quotient ¡ ¤ G Z is called the projective general linear group, and denoted PGLn £ q . ¨ ¤ ¤ ¤ £ £ Also, since det £ AB det A det B , the determinant map is a group homomor- ¤ phism from GLn £ q onto the multiplicative group of the field, so its kernel is a normal ¤ £ subgroup of index q ¦ 1. This kernel is called the special linear group SLn q , and ¤ consists of all the matrices of determinant 1. Similarly, we can quotient SLn £ q by the ¤ subgroup of scalars it contains, to obtain the projective special linear group PSLn £ q , ¤ sometimes abbreviated to Ln £ q . Now an invertible matrix takes a basis to a basis, and is determined by the image of an ordered basis. The only condition on this image is that the ith vector is linearly independent of the previous ones—but these span a space of dimension i ¦ 1, which ¤ ¤ ¤¥¤¦¤¦¤ ¤ i 1 ¤ n n n 2 n n 1 £ £ ¦ £ ¦ £ ¦ £ ¦ ¡ has q ¡ vectors in it, so the order of GLn q is q 1 q q q q q q , ¤ ¤¥¤¦¤¦¤ ¤ ¨ © n § n 1 2 2 n £ ¦ £ ¦ £ ¦ which can also be expressed as q ¡ q 1 q 1 q 1 . ¤ ¤ ¤ £ £ ¦ The orders of SLn £ q and PGLn q are equal, being GLn q divided by q 1.
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