FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 49 Figure 4. The Fano plane. 11. Projective Space One can approach the study of projective spaces from a number of different angles. Since we will be primarily interested in projective spaces of finite dimension over finite fields, our spaces themselves are finite and so fall most naturally it seems to me! into the sphere of combinatorics. Our approach is, therefore, combinatorial. 11.1. Incidence structures. "e define a category IncStruct as follows. Objects# An object in IncStruct is a finite tuple P1,...,P�,I! whereP 1,...,P� are sets, andI isa subset ofP P . 1 × · · · × � Some terminology# An object #% P 1,...,P�,I! is called an incidence structure. &lements from setP i are said to have typei. IfI containsI an element p , . , p ! then, for 1 i,j �, we say thatp is incident 1 � ≤ ≤ i withp j. The incidence structure is called finite ifP 1,...,P� are all finite. To complete the definition of this category we must, of course, define what the arrows are. "e will do this shortly, but first some e(amples. Note that the definition of an incidence structure is extremely general* in most cases incidence structures are only studied subject to extra a(ioms. Example 22. "e have seen the category SimpleGraph earlier in lectures, and we con+ sidered the category Graph in exercises. Both of these categories could be seen as (full! subcategories of IncStruct. In particular we define the category Graph as follows: Objects# An object #% P ,P ,I! is an incidence structure with - types. (Elements G 1 2 ofP 1 are what we thin. of as vertices, elements ofP 2 are edges!. "e require that, for any p P , 2 ∈ 2 p P p , p ! I -. |{ 1 ∈ 1 | 1 2 ∈ }| ≤ Since Graph is a full subcategory of IncStruct, the definition of arrows in Graph is the same as the definition in IncStruct, and we give this definition in a moment. Example 23. An abstract projective plane is an incidence structure P,L,I! such that 1) any two elements ofP are incident with a uni/ue element ofL* -! any two elements ofL are incident with a uni/ue element ofP. 0! there are four elements ofP such that no element ofL is incident with more than two of them. "e call elements ofP points and elements ofL lines. The third axiom listed above 1presence of a /uadrangle2! is there simply to eliminate some degenerate examples. IfP is finite it is easy to see thatL is also finite and, in fact, that P % L . The smallest finite projective plane has P % 3. It is called the Fano plane and is| represented| | | in Figure 4* in this representation elements| | ofP are drawn as points, and elements ofL are drawn as lines (in si( cases) or as a circle; each element ofL is incident with three elements ofP and vice versa!. Example 24. 5etV be a vector space of dimensionn over a fieldk. "e define projective space 67 V !, or 67n 1 k!, to be the incidence structure V1,...,Vn 1,I! where, fori%1, . , n 1, − − − Vi is the set of subspaces ofV of dimensioni and I #% v 1, . , vn 1! V 1 V n 1 v 1 < v2 < <v n 1 . { − ∈ × · · · × − | ··· − } 50 NICK GILL In other words two subspaces are incident if and only if one is contained in the other. This fact allows us to rela( language a little: we say things li.e 1v1 lies onv 22, or 1v2 containsv 12 when we really mean that 1v1 andv 2 are incident2. $ subspaceU V is said to have projective dimension, pdim U!, equal toi 1 and we ∈ i − call elements ofV 1 1points’, elements ofV 2 1lines2, and elements ofV n 1 1hyperplanes’. Ifk is − finite of orderq, then we sometimes write 67 n 1 q! for 67n 1 k!. The incidence structure − − 672 q! is called the Desarguesian projective plane of orderq. (E11.1) Show thatPG 2(2) and the Fano plane are the same incidence structure. (We would do better to write that “PG2(2) and the Fano plane are isomorphic as incidence structures”, but we have not yet defined what we mean by isomorphism.) (E11.2*) Show that, for any prime powerq,PG 2(q) is an abstract projective plane. 11.-. Some counting. "e are interested in calculating the order ofV 1,...,Vn 1 in 67n 1 q!. To do this it is convenient to introduce Gaussian coefficients. 5etq be a prime power,m and−n positive− integers. Then define n n n m 1 n q 1! q q! q q − ! 10! #% − − ··· − . m qm 1! qm q! q m q m 1! � �q − − ··· − − n Lemma 11.1. 1! The number of subspaces of projective dimensionm 1 in 67 n 1 q! is . In − m q n qn 1 − particular, the number of points in 67n 1 q! is % − . − 1 q q 1 � � -! The number of subspaces of projective dimensionm 1 containing− a subspace of projective dimension n l � −� l 1 in 67 q! is − . n 1 m l q − − − Proof. 5etV beann-dimensional� � vector space overk%F q. To prove a! we count the number of linearly independentm-tuples of vectors inV . The first entry in them+tuple can be chosen to be any non-zero vector, there areq n 1 of these; the second must lie outside the span of the first, so there areq n q choices for this, thenq n q−2 for the third and so on. "e conclude that the numerator of the right-hand− side of 10! corresponds− to the number of linearly independentm-tuples of vectors. Now the result is completed by observing using the same reasoning! that the denominator of the right+hand side of 10! corresponds to the number of linearly independentm+tuples of vectors all lying inside any givenm-dimensional subspace ofV. (E11.3)Prove (b). � Gaussian coefficients have properties resembling those of binomial coefficients, to which they tend as q 1. → n n (E11.4)Prove that limq 1 = . → m q m (E11.5)Prove that � � � � n n m+1 n n+1 +q − = . m m 1 m � �q � − �q � �q 11.0. Collineations and the fundamental theorem. "e are ready to complete the definition of the category IncStruct# 1 1 1 2 2 2 F !rro"s# 5et 1 % P1 ,...,P� ,I !, 2 % P1 ,...,P� ,I ! be incidence structures. An arrow 1 2 is a set of� functionsIφ #P 1 P 2 such thatI I → I i i → i p , . , p ! I 1 % φ p !,...,φ p !! I 2. 1 � ∈ ⇒ 1 � ∈ Note that we only define arrows between incidence structures having the same number of types. F 1 2 The notion of isomorphism is now clear: An isomorphism 1 2 is a set of� bijectionsφ i #P i P i such that I → I → p , . , p ! I 1 φ p !,...,φ p !! I 2. 1 � ∈ ⇐⇒ 1 � ∈ FINITE PERMUTATION GROUPS AND FINITE CLASSICAL GROUPS 51 One would expect that an isomorphism 1 F 1 would be .nown as an automorphism, however the termi+ nology for this category is a little different#I → such I a thing is called a collineation. As usual we denote the set of all collineations of an incidence structure by Aut( ). I I Example 25. LetV beann+dimensional vector space over a fieldk. The action of ;L V! on the setV extends naturally to an action on the set of subspaces ofV# ifv 1, . , vk V andg ;5 V !, then we writeU% v , . , v and define ∈ ∈ � 1 n� U g % v g, . , vg . � 1 k� (E11.6*)Prove that this action is well-defined, and that the action preserves the incidence relation for PG(V). This exercise implies that we have a homomorphismφ # ;5 V! Aut 67 V !). The next result asserts that this homomorphism is also surjective, i.e. all collineations→ of 67 V ! are induced by a semilinear transformation. Theorem 11.2. (Fundamental theorem of projective geometry) If dimV 0, then Im φ! % $ut 67 V!!. ≥ Proof. This is omitted. See <Camb, Chapter 1] or <Tay9->. � The first isomorphism theorem of group theory implies, then, that Aut 67 V!! ∼% ;5 V!/ .er φ!. (E11.7) Prove that ker(φ)= αI GL(V) α k . { ∈ | ∈ } 5et us writeK for .er φ! and observe thatK is just the group of scalar transformations in ;L V ). Note thatK is actually a subgroup of 75 V!. "e now define three new groups in terms of this subgroupK. 1) 6;5 V ! #% ;5 V!/K* -! 675 V ! #% 75 V!/K* 0! 6S5 V!#%S5 V!/ K S5 V !). ∩ ForX ;, G, S we write 6XL n k! as a synonym for 6XL V!. Observe∈{ that} the Fundamental theorem of projective geometry could be e(pressed as follows: If dimV 0, then ≥ 6;5 k! % Aut 67 V!. In particular the three groups just defined all act faithfully on 67 V!. (E11.8)Prove thatK is central in GL(V). "an you characteri#e those fieldsk and those vector spacesV for whichK is central in ΓL(V)$ (E11.9*)Prove that 1, ifn is odd; PGLn(R) : PSLn(R) = | | 2, ifn is even. � 11.4. %ualit&. Next we define a variant of the category IncStruct that we call WIncStruct. The objects are the same @ incidence structures @ but we allow more arrows# 1 1 1 2 2 2 F !rro"s# 5et 1 % P1 ,...,P� ,I !, 2 % P1 ,...,P� ,I ! be incidence structures. An arrow 1 2 in WIncStruct isI a permutationπ S ,I and set of� functionsφ #P 1 P 2 such that I → I ∈ k i i → iπ 1 2 p , .