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On the proportion of derangements in cosets of primitive permutation groups
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On the proportion of derangements in cosets of primitive permutation groups
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ON THE PROPORTION OF DERANGEMENTS IN COSETS OF PRIMITIVE PERMUTATION GROUPS by Andrei Bogdan Pavelescu A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MATHEMATICS) May 2012 Copyright 2012 Andrei Bogdan Pavelescu Dedication To Elena, Maria and Ion. ii Acknowledgments I would like to thank my advisor, Robert Guralnick, for his guidance and support. Were it not for his patience, insight and generosity, my thesis would not exist. Thank you! I would also like to thank professors Francis Bonahon, Eric Friedlander, Jason Fulman, Thomas Geisser, Larry Goldstein, Susan Montgomery and Pham Huu Tiep for the knowl- edge and support they have so generously offered. I want to thank Taylan Bilal, Laura Chakerian, Dmitri Chebotarev, Yang Huang, Mi- haela Ignatova, David Karapetyan, Vlad Vicol, Xin Wang and Youngyun Yun; graduate school would have been bleak without your friendship. A special thank you goes to my family and especially to my parents. I thank you for providing me with all your love and support. Finally, I want to thank my spouse, Elena Pavelescu. Through these years you were a constant source of love and inspiration; you are my role model. iii Table of Contents Dedication ii Acknowledgments iii Abstract v Chapter 1 Introduction 1 Chapter 2 Background 5 2.1 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Primitive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Burnside’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Actions onk-tuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Wreath Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 (Semi)Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . 10 2.7 Frobenius Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.8 The Aschbacher–O’Nan–Scott Theorem . . . . . . . . . . . . . . . . . . . 13 Chapter 3 Machinery 15 3.1 Counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Chapter 4 The Affine Case 22 Chapter 5 Regular Normal Nonabelian Subgroup 25 References 28 iv Abstract Motivated by questions arising in connection with branched coverings of connected smooth projective curves over finite fields, I am studying the proportion of fixed point free elements (derangements) in cosets of normal subgroups of primitive permutations groups. Using the Aschbacher–O’Nan–Scott theorem for primitive groups to partition the problem, I provide complete answers for affine groups and groups which contain a regular normal nonabelian subgroup. v Chapter 1 Introduction The study of the fixed points of permutations has a long history, starting with a probabil- ity theorem of Montmort [M] (1708) who proved that the average number of permutations of sizen withk fixed points tends to 1=(ek!). Further connections with probability theory come in relation with card shuffling. In a more algebraic context, letA be a transitive permutation group acting on a setX with n elements. Let S 0 denote the set of fixed point free elements ofG. By a classical result of C. Jordan [Jo],S 0 is nonempty. Motivated by number theoretic applications such as the number field sieve, H.W. Lenstra, Jr. [BLP] (1990) asked for a lower bound for s 0 := jS 0 j jAj : Cameron and Cohen [CC] (1992) proved thats 0 1=n with equality if and only ifA is a Frobenius group of ordern(n 1), withn a prime power. In 1997 Guralnick and Wan [GW] proved that ifs 0 > 1=n, then the next bound iss 0 = 2=n, with equality for a Frobe- nius group of ordern(n 1)=2 withn an odd prime power,Z=3Z orA 5 . The question of studyings 0 arises in a more arithmetic setting. LetF q be a finite field of characteristic p and let f(T )2 F q [T ] be a polynomial of degree n > 1 which is not a polynomial in T p . S. Chowla [Ch] asked for an estimation 1 ofV f :=jf(F q )j. A result of Birch and Swinnerton-Dyer [BS] shows that, provided the Galois group off(T )t = 0 overF q (t) isS n , then: V f = n X k=1 (1) k1 k! ! q +O( p q); where the constant term in the error does not depend on f, only on n. Unless f is a permutation polynomial (V f =q),V f <q and a known elementary upper bound is V f q q 1 n : When interested in asymptotic upper bounds, it turns out they do depend onf. LetA be the Galois group off(T )t = 0 overF q (t) and letG be the Galois group of f(T )t = 0 overF q (t). Both groups act transitively on the set ofn roots off(T )t = 0. Furthermore, the geometric monodromy group G is a normal subgroup of the arithmetic monodromy group A, with A=G cyclic generated by xG. If S 0 denotes the set of fixed point free elements inxG, then the Cebotarev density theorem for function fields yields: V f = (1 jS 0 j jGj )q +O( p q); with the constant error term depending only onn. Therefore the problem is reduced to the understanding ofs 0 =jS 0 j=jGj. Unlessf is an exceptional polynomial (it induces bijections in arbitrarily large degree extensions ofF q ),s 0 > 0 with the next bounds 0 = 1=n holding ifA = G is a Frobenius group of ordern(n 1) withn a prime power (Lenstra). Not surprisingly, the next bound fors 0 is 2=n as given in: 2 Theorem 1.0.1 (Guralnick–Wan, 1997). Let f(T ) be a polynomial overF q of degree n> 6 which is not a polynomial inT p . Ifs 0 > 1=n, thens 0 2=n with equality holding iffA = G is a Frobenius group of ordern(n 1)=2 with n a prime power. In particular, V f (1 2=n)q +O n ( p q) unlessf is exceptional orA =G is a Frobenius group of order n(n 1). The proof of this theorem uses the classification of finite simple groups. If the degree off is not divisible by the characteristic ofF q , which is the same as saying f as seen as a morphism fromP 1 toP 1 has tame ramification at1, thens 0 > 1=6 [GW] whenevers 0 > 0 . Thus eitherf is bijective orV f (5=6)q +O n ( p q). If all ramification is tame, then Guralnick and Wan [GW] proved thats 0 > 0 implies s 0 16=63 and the bound is the best possible. In [GW], Guralnick and Wan generalized these results to branched coverings of smooth projective curves defined over a finite field. Reducing to the case where the covering is indecomposable (the corresponding arithmetic monodromy group is primitive), the authors concluded: Theorem 1.0.2 (Guralnick–Wan). Let : X! Y be a separable branched covering of degree n with X;Y; defined overF q . Assume that one of the branch points is totally ramified and isF q -rational. Let p be the characterisitic ofF q . Let A be the arithmetic monodromy group of the covering andG the geometric monodromy group. Then one of the following holds: (a) r 2 = 0,s 0 = 0 and the covering is exceptional; 3 (b) r 2 = 1,s 0 = 1=n andA =G is Frobenius of ordern(n 1) withn a prime orp a ; (c) r 2 = 2,s 0 = 2=n andA =G is Frobenius of ordern(n 1)=2 withn an odd prime orp a (withp> 2); (d) s 0 > 2=n; or (e) n 6,A =G and 1=ns 0 2=n orn = 4,jA=Gj = 2 ands 0 = 2=4. In the same paper, the authors commented that “...there should be a version of the pre- vious result without the assumption that we are dealing with monodromy groups of poly- nomials (or more generally coverings with a totally ramified rational point)”. In my thesis, I study this situation and provide answers for the affine case and the regular nonabelian normal subgroup case. 4 Chapter 2 Background 2.1 Group Actions The definitions and notions presented in this chapter are based on [DM] and [Wi]. For convenience, I have chosen to work with left actions, but everything holds, mutatis mutan- dis, for right actions. Let G be a group and let be a nonempty set. An action of G on is a function G ! given by (x;)!x() with the following properties: (i) 1() =, for all2 and 1 the identity element ofG. (ii)y(x()) =yx() for all2 andx;y2G. Example 2.1.1. IfSym( ) denotes the set of all bijections : ! , thenSym( ) acts naturally on by (;)!(). The above example gives rise to the following construction. To each elementx2 G associate x 2 Sym( ) by x () = x(). This defines a map : G ! Sym( ), (x) = x ; is well defined as x x 1 = x 1 x = 1 . Furthermore, the properties of the group action show that is a group homomorphism. In general, any homomorphism of G intoSym( ) is called a permutation representation ofG on . Any group action gives 5 rise to a permutation representation and any permutation representation induces a group action. The degree of an action (or the corresponding permutation representation) is the size of . We call an action faithful if the kernel of the associated permutation representation is trivial. To say a groupG acts faithfully on a set is the same as saying thatG embeds into Sym( ). Any such group is called a permutation group. For2 , letG() :=fx() : x2 Gg and call this set the orbit of. LetG := fx2G :x() =g and call this set the (point) stabilizer of inG. Then we have: Theorem 2.1.2. Suppose thatG is a group acting on a set and thatx;y2G and;2 . Then: (i) Two orbitsG() andG() are either equal or disjoint, so they form a partition of . (ii) The stabilizer G is a subgroup of G and G = xG x 1 whenever x() = . Moreoverx() =y(),xG =yG : (iii) (The orbit-stabilizer property)jG()j =jG :G j for all2 . IfG is finite, then jG()jjG j =jGj. A groupG acting on a set is said to be transitive on if it has only one orbit, and so G() = for all 2 . A group which is not transitive is called intransitive. A transitive groupG is called regular ifG = 1 for all2 . The previous theorem yields this following result. Corollary 2.1.3. SupposeG acts transitively on . Then (i) The stabilizersG (2 ) form a single conjugacy class. (ii) For every2 ,jG :G j =j j. (iii) IfG is finite, thenG is regular,jGj =j j. 6 2.2 Primitive Groups LetG be a group acting transitively on a set . A nonempty subset of is called a block for the action of G if for all x2 G either x() = or x()\ =;. Any block of size greater than 1 is called nontrivial. G is called primitive if it has no nontrivial blocks; otherwise it is called imprimitive. If G acts trasitively on and 2 , there is a Galois like correspondence between the set of blocks containing and the lattice of subgroups ofG that containG . This fact yields: Corollary 2.2.1. LetG be a group acting transitively on a set with at least two points. ThenG is primitive, each point stabilizerG is a maximal subgroup of G. One obvious fact deriving from this is that a regular permutation group is primitive if and only if it has prime degree. 2.3 Burnside’s Lemma LetG be a finite group acting on a finite set . For everyx2G denote byFix(x) =f2 :x() =g. The following result is usually refered to as the “Burnside Lemma”: Theorem 2.3.1. (Cauchy–Frobenius) LetG be a finite group acting on a finite set . Ifm is the number of orbits of this action, then mjGj = X x2G jFix(x)j: Proof. LetF :=f(x;)2 G : x() = g: Assume the orbits of the action are 7 1 ; 2 ;:::; m . On one hand jFj = X x2G jFix(x)j: On the other hand, using the orbit-stabilizer property, we get: jFj = m X i=1 X 2 i jG j = m X i=1 X 2 i jGj j i j = m X i=1 jGj =mjGj: 2.4 Actions onk-tuples LetG be a group acting on a set and let k denote thek-th cartesian power of . The original action ofG on induces a component-wise action on k . Furthermore, the subset of k consisting ofk-tuples of distinct points isG-invariant for any choice ofG andk. We denote this set by (k) . If a groupG is acting on a set andk is an integer with 1kj j, then we sayG is k-transitive ifG is transitive on (k) . We say thatG is highly transitive if is infinite and G isk-transitive for all integersk 1. Directly from the definitions one can derive the following useful results. Proposition 2.4.1. IfG is a finitek-transitive group, thenjGj is divisible byn(n1):::(n k + 1). Proposition 2.4.2. If G is a k-transitive group with k 2 and N is a nontrivial normal subgroup ofG, thenN is (k 1)-transitive. 8 2.5 Wreath Products The standard examples of imprimitive subgroups of S n are wreath products. Based on their inner structure, wreath products are fertle generators of inductive counting arguments. While one can define the wreath product of two groups in full generality, for the purpose of this paper we are going to restrict our attention to permutation groups. Let n = km and consider the semidirect product S m k oS m acting on a partition of f1; 2;:::;ng intom blocks of sizek; eachS k acts as the full group of permutations of its corresponding block and the wreathing action of S m permutes the m orbits. Denote this product byS k oS m , the wreath ofS k withS m . A list of all the transitive imprimitive maxi- mal subgroups ofS n is given by all theS k oS m withn =km;k> 1;m> 1. Although this construction is imprimitive on a set ofn elements, it can still act prim- itively on different sets. The following is an example of a primitive action of a wreath product: Example 2.5.1. (The product action) Letn = k m ;k > 2 andm > 1. Let = m 1 with j 1 j =k. Let the elements ofS m k act component-wise on the poits of , i.e. ( 1 ; 2 ;:::; m )(a 1 ;a 2 ;:::;a m ) = ( 1 (a 1 ); 2 (a 2 );:::; m (a m )): The wreathing action ofS m permutes the coordinates on them-tuples by : (a 1 ;a 2 ;:::;a m )! (a (1) ;a (2) ;:::;a (m) ): The wreath product occurs in one more instance of a primitive action. 9 Example 2.5.2. (The diagonal type) LetT be a nonabelian simple group and letOut(T ) denote the group of outer automorphisms. Consider the construction T k (Out(T )S k ) = (ToS k )Out(T ): The normal subgroupToS k is extended by the group of automorphisms acting “diagonally” (in the same way) on allk copies ofT . This group contains a subgroup ofAut(T )S k con- sisting of a diagonal copy ofT (f(t;t;:::;t) :t2Tg) extended by its outer automorphism group and the permutation group. This subgroup has indexjTj k1 , so the permutation of the group on the cosets of this subgroup gives an embedding of the whole group intoS n , wheren =jTj k1 . 2.6 (Semi)Linear Transformations LetF be a field andd be a positive integer. LetAG d (F ) denote thed-dimensional affine geometry over F (the collection of all affine subspaces of F d ). An automorphism of the affine geometry is a permutation on the set of points ofF d which maps each affine subspace to an affine subspace of the same dimension. Example 2.6.1. Leta2 GL d (F ) . For any vectorv2 F d , define a transformationt a;v : F d !F d by t a;v :u!au +v: Each of this maps is an automorphism of AG d (F ). The set of all of these maps (for all a2 GL d (F ), v 2 F d ) forms the affine group AGL d (F ) of dimension d over F . This group is 2–transitive; it’s also a split extension ofT byGL d (F ), where T =ft 1;v :v2F d g 10 is a regular normal subgroup isomorphic toF d (the group of translations). Example 2.6.2. To determine the full automorphism group of the affine geometry, one needs to consider the group of field automorphisms ofF ,Aut(F ). For each2 Aut(F ) there is a permutation ofF d ,t , defined by t ([u 1 ;u 2 ;:::;u d ] t ) := [(u 1 );(u 2 );:::;(u d )] t : These mappings form a subgroup ofSym(F d ) isomorphic toAut(F ). This subgroup to- gether withAGL d (F ) generates the groupAL d (F ) of affine semilinear transformations. The elements ofAL d (F ) are permutations ofF d of the form t a;v; :u!at (u) +v; wherea2 GL d (F ),v2 F d and2 Aut(F ). Ford 2,AL d (F ) is the full automor- phism group ofAG d (F ). In the cases whereAut(F ) = 1 (for instance ifjFj is prime or F =R orQ) we haveAL d (F ) =AGL d (F ). One important subgroup ofAGL d (F ) is the affine special linear group ASL d (F ) :=ft a;v 2AGL d (F ) :det(a) = 1g: It still contains the translations as a normal regular subgroup and the stabilizer of a point is isomorphic toSL d (F ). The following is a useful result. Proposition 2.6.3. LetF be a field and letd 2. Then: (i)ASL d (F ) is a 2–transitive group. (ii)ASL d (2) :=ASL d (F 2 ) =AGL d (F 2 ) is a 3–transitive group. 11 2.7 Frobenius Groups A Frobenius group is a transitive permutation group which is not regular, but in which only the identity has more than one fixed point. For a finite Frobenius group, the set of fixed point free elements plays an important role. Theorem 2.7.1. (Structure Theorem for Finite Frobenius Groups) LetG be a finite Frobe- nius group, letG be a point stabilizer and let K :=fx2G :x = 1orFix(x) =;g: (i)K is a regular normal subgroup ofG. (ii) For each odd primep, the Sylowp-subgroups ofG are cyclic, and the Sylow 2- subgroups are either cyclic or quaternion. If G is not solvable, then it has exactly one nonabelian composition factor, namelyA 5 . (iii)K is a nilpotent group. Example 2.7.2. AGL 1 (F ) is a Frobenius group withK =F andG =F . Example 2.7.3. The dihedral groupsD 2n are all Frobenius groups. Assume the Frobenius groupG is acting on a finite set . For an element2 ,G acts regularly on each of his orbits on nfg. It follows thatjGj = nd, withd dividing (n 1). In particular,G is quite a small subgroup ofS n . Ifd = n 1, then any two nonidentity elements ofK are conjugate inG, thusK is an elementary abelianp-group and thusn =p a for some primep. Ifd = (n 1)=2, there are two conjugacy classes (inG) of nonidentity elements ofK. It then follows thatK is an elementary abelianp-group for some odd primep. 12 2.8 The Aschbacher–O’Nan–Scott Theorem A minimal normal subgroup of a nontrivial groupG is a normal subgroup 16= K which does not properly contain any other nontrivial normal subgroup ofG. The socle of a group G is the subgroup generated by the set of all minimal normal subgroups ofG; it is denoted bysoc(G) and, by convention,soc(G) = 1 ifG has no minimal normal subgroups. Any nontrivial finite group has a nontrivial socle. The following facts follow from the defini- tions. Proposition 2.8.1. The socle is a characteristic subgroup ofG. Proposition 2.8.2. Any two distinct minimal normal subgroups commute. Proposition 2.8.3. A primitive group contains at most two minimal normal subgroups. We now have everything we need in order to give the classification of primitive per- mutation groups. It shall prove an essential tool in our efforts to study the proportion of derangements. Theorem 2.8.4. (Aschbacher–O’Nan–Scott Theorem) LetG be a finite primitive group of degreen, and letH be the socle of G. Then either (a)H is a regular elementary abelianp-group for some primep,n =p m :=jHj, andG is isomorphic to a subgroup of the affine groupAGL m (p); or (b)H is isomorphic to a direct powerT m of a nonabelian simple groupT and one of the following holds: (i)m 6,H is regular andn =jTj m ; (ii)m 2 andG is a group of “diagonal type” withn =jTj m1 ; 13 (iii)m 2 and for some proper divisord ofm and some primitive subgroupU with socle isomorphic toT d ,G is isomorphic to a subgroup of the wreath productUoS m=d with the “product action”, andn =l m=d wherel is the degree of U; (iv)m = 1 andG is isomorphic to a subgroup ofAut(T ). 14 Chapter 3 Machinery 3.1 Counting LetA be a permutation group acting (fatihfully) on a setX which hasn elements. Denote by( 2) the minimal number of elements ofX moved by a nonidentity element ofA. Lemma 3.1.1. The number of orbits ofA acting onX is less or equal ton 2 . Proof. Ifr is the number of orbits ofX under the action ofA, then, by Burnside’s Lemma rjAj = X x2A jFix(x)j =n + X x6=1 jFix(x)jn + (jAj 1)(n) =jAjn (jAj 1)) )rn jAj 1 jAj n 2 : Remark 3.1.2. Notice that the above inequality is strict unlessjAj = 2. LetG be a normal subgroup ofA such thatA=G is cyclic, generated byx. Lemma 3.1.3. Letr =r(X) be the number of common (A;G) orbits onX. Then 1 jGj X g2xG jFix(g)j =r: 15 Proof. Without loss of generality, one may assumeA is transitive. (The orbits of theA- action form a partition ofX which has as a subpartition the orbits of theG-action). Claim:G is transitive iff there existsg2G such thatxg has a fixed point. Proof of claim: ")" Let 2 X. Since G is transitive,9g 2 G such that g() = x 1 ())xg() =. "(" Letg2 G,2 X such thatxg() = ) g 1 x 1 () = ) xg 1 x 1 () = x()) g 1 () = x() for someg 1 2 G, asG is a normal subgroup. If is an arbitrary element of X, since A is transitive, there exists h2 A such that = h(). Under the current assumptions onA andG, there existt2N andg 2 2G such thath =x t g 2 . It then follows that =x t g 2 () =x t1 xg 2 x 1 x() =x t1 g 3 () =::: =x ti g i+2 () =::: =g t+2 () where inductivelyg i+1 :=xg i x 1 g 1 2G, sinceG is normal inA. Since was arbitrary, it follows thatG is transitive. By the claim, ifG is not transitive, both sides of the equation are 0. So we assumeG is transitive (r=1). Set Y =f(xg;!)2xGXjxg(!) =!g; nonempty by the claim. Let A ! and G ! denote the corresponding point stabilizers. If 16 xg(!) =!; thenA ! \xG =xgG ! , thusjA ! \xGj =jG ! j: One has X g2xG jFix(g)j =jYj = X !2X jA ! \xGj = X !2X jG ! j = X !2X jGj jXj =jGj; sinceG was assumed transitive. From this point on, unless otherwise specified,A andG are assumed transitive. For all 0 i n, define S i :=fg2 xG :jFix(g)j = ig: Let s i :=jS i j=jGj. Let r k denote the number of common (A;G)–orbits of the component-wise actions onX (k) , thek-fold cartesian product with all diagonals removed. Lemma 3.1.4. The following are equivalent: (a) r 2 = 0; (b) s 0 = 0; (c) every element in the cosetxG fixes a unique point; (d) every element in the cosetxG fixes at most one point; (e) every element in the cosetxG fixes at least one point. Proof. Since A and G are transitive, by Lemma 3.1.3 it follows that (c),(d) and (e) are equivalent. Also, by the definition ofs 0 , (b) and (e) are equivalent. Fora;b2 X,a6= b we have A(a;b) = G(a;b) if and only if there exists g 2 G such that xg 2 A a \A b . Thusr 2 6= 0 is equivalent to some element inxG fixing at least two points; therefore (a) is equivalent to (d). A triple (A;G;X) with the above properties is called exceptional. 17 3.2 Combinatorics Lemma 3.2.1. Assumingr 2 1, we get s 0 r 2 n + (n 2)r 2 r 3 n(n) : Proof. SinceA andG both act transitively, it follows thatr 1 = 1. Furthermore, s 0 +s 1 +s 2 +::: +s n = 1: (3.2.1) By Lemma 3.1.3 applied to (A;G;X (k) ), for 1kn, we get r k = 1 jGj X g2xG jFix(g)j = 1 jGj n X i=0 X g2S i jFix(g)j = 1 jGj n X i=k jS i jP k i = n X i=k jS i j jGj i k k!; which yields n X i=k i k s i = r k k! ; 1kn: (3.2.2) Remark 3.2.2. The sums actually go up ton ass n =s n1 =::: =s n+1 = 0. Subtracting 3.2.1 from the first equation of 3.2.2, one gets s 0 = n X i=2 i 1 1 s i : (3.2.3) 18 By multiplying 3.2.3 by n 2 and subtracting the third equation of 3.2.2, we get ns 0 2 r 2 2 = n X i=2 (ni)(i 1) 2 s i 0: (3.2.4) The last formula immediately implies that s 0 r 2 =n, with equality if and only if s 2 = s 3 =::: = 0. Sincer 2 1, there exists (a;b)2X (2) such thatA(a;b) =G(a;b). But then, there existsg2 G such thatx 1 (a;b) = g(a;b)) xg(a;b) = (a;b)) xg2 xG\A a;b ; this can only happen ifx = g 1 ass 2 = s 3 = ::: = 0, thusA = G is a Frobenius group. UnlessjAj =n(n 1) orjAj =n(n 1)=2,s 0 = (n 1)=jGj> 2=n. The stabilizer of a point acts as fixed point free automorphisms of the regular normal subgroupN. Thus, by considering nontrivial conjugacy classes, it follows thatN is ap-elementary group withp prime. Thusn =p a withp odd ifjAj =n(n 1)=2. Multiplying the second equation of 3.2.2 by n2 3 and subtracting the third equation of 3.2.2, it follows that (n 2)r 2 r 3 6 = n X i=2 (ni)i(i 1) 3! s i 0: (3.2.5) Finally, by multiplying 3.2.4 by n 3 and subtracting 3.2.5, one gets n 3 ( ns 0 2 r 2 2 ) = n1 X i=2 (ni)(ni)(i 1) 3! s i 0; thus s 0 r 2 n + (n 2)r 2 r 3 n(n) : (3.2.6) 19 By 3.2.5, ifr 2 2, thens 0 2 n . In a similiar setting, Guralnick and Wan [GW] proved that it suffices to study the case where A is primitive. For the rest of this paper we shall assume r 2 = 1 and A primi- tive. Note that this impliesG is transitive since otherwise the orbits ofG would constitute imprimitivity blocks forA. The bound in 3.2.6 reduces to s 0 1 n + n (r 3 + 2) n(n) ; (3.2.7) which immediately implies the following lemma. Lemma 3.2.3. Ifr 3 + 2<, thens 0 > 2 n . Let (a;b) be a representative of the common (A;G) orbit on X (2) . Let A a;b be the stabilizer ofa andb acting onX and letr denote the number of orbits of this action. Proposition 3.2.4. r 3 + 2rn 2 . Proof. First notice that if (a;b;c)2 X (3) such thatA(a;b;c) = G(a;b;c), then definitely A(a;b) =G(a;b) and thus for everyi = 1; 2;:::;r 3 , there existsc i 2X such that (a;b;c i )2 O i , whereO 1 ;O 2 ;:::;O r 3 are the common (A;G) orbits onX (3) . Denote byfo 1 ;:::o r g the collection ofA a;b orbits and define a set map from' :fO 1 ;O 2 ;:::;O r 3 g!fo 1 ;:::o r g by '(A(a;b;c i )) =A a;b c i . Then '(A(a;b;c i )) ='(A(a;b;c j )),A a;b c i =A a;b c j ,9g2A;g(a;b;c i ) = (a;b;c j ); which is to say, ' is a well-defined injection. Sincefag;fbg = 2 Im('), it follows that r 3 + 2rn 2 , by Lemma 3.1.1. 20 The above result, 3.2.3 and the remark following Lemma 3.1.1 imply the following lemma. Lemma 3.2.5. a) If> 2n 3 , thens 0 > 2 n ; b) If = 2n 3 , thens 0 > 2 n unlessA a;b is a subgroup of order 2. At this point, we need more information about. It turns out that if the groupA is affine, the extra geometric structure is sufficient to fully classify all possibilities in this case. Under the assumption thatr 2 = 1, we can derive an upper bound fors 0 : s 0 n X i=0 s i i 0 i 1 + i 2 = 1r 1 + r 2 2 = 1 2 : Remark 3.2.6. The equality in the above inequality holds if and only ifs 3 = s 4 = ::: = s n = 0. 21 Chapter 4 The Affine Case From this point on we are going to assume that A AL d (q) is affine, acting on a d- dimensionalF q -vector spaceV , withn = q d . One can identifyV as a subgroup ofA (as translations) andA =VA 0 . The isotropy groupA 0 (the point stabilizer of 0) is a subgroup of L d (q), the group of semilinear transformations. The following result follows from Lang’s Theorem ([GW], Lemma 2.3) Lemma 4.0.7. a) Ifd> 1, then (q1)n q . b) Ifd = 1 andq is prime, then =q 1. c) Ifd = 1 andq =q e 0 withe prime and minimal, thenqq 0 . By Lemma 3.2.5 and Lemma 4.0.7 it follows thats 0 > 2=n unlessq 3 orn = 4 or 9. Ifn = 4, thenA = A 4 orS 4 . Since the Klein 4–group is not 2–transitive andS 4 andA 4 are, this implies that 1. (A;G;V ) is exceptional or 2.A =S 4 ,G =A 4 ,s 0 = 1 2 or 3.A =G =A 4 ,A is Frobenius ands 0 = 1 4 . Ifn = 9, then eitherA =G is Frobenius, (A;G;V ) is exceptional, ors 0 1 3 . 22 Ifq = 3,n = 3 d > 9 by Lemmas 3.2.5 and 4.0.7 it follows that eitherA is Frobenius, s 0 > 2=n orx acts as a reflection andA 0;v has order 2 for any nonzerov fixed byx. Let W denote the hyperplane fixed byx. For any suchv,x t g(0;v) = x t gx t (0;v)2 G(0;v) shows thatA(0;v) =G(0;v). Asr 2 = 1, for any two nonzero distinct elementsw;w 0 2W , (0;w) and (0;w 0 ) are contained in the common (A;G)–orbit. It follows that all nonzero elements ofW are contained in the sameA 0 –orbit. In particulara(v) =w fora2A 0 and 06=w2W shows thata 1 xa(v) =v anda 1 xa(0) = 0. SinceA 0;v =f1;xg, the central- izer ofx acts transitively on all nonzero elements ofW . Letu be a vector in the eigenspace of1. As A 0 is irreducible on V (otherwise the nontrivial A 0 –invariant subspace of V would constitute an imprimitivity block for A),A 0 u = A 0 v. For somea2 A 0 ,a(u) = v, thusu =a 1 xa(u), so this means some reflectionx 0 6=x centralizesu. Asd> 2,x andx 0 both fix some nonzero vectorw inW . ThenA 0;w has order greater than 2 and so doesA 0;v asw andv are in the sameA 0 –orbit. But this is a contradiction. In the casen = 2 d > 4, we may assumex fixes 0. IfxG does not contain a transvection (unipotent element fixing a hyperplane), then 3n=4 and thus, by Lemma 3.2.5,s 0 > 2 n . So, without loss of generality, we may assumex is a transvection. As above, ifW is the fixed hyperplane ofx, asr 2 = 1, all the nonzero vectors ofW are in the sameA 0 –orbit. LetH be the subgroup ofA 0 generated by transvections. As all the nonzero elements ofW are in the sameA 0 –orbit, for eachw2 Wnf0g, there is a transvection w centered atw. This leavesW as the only candidate for a nontrivial invariant subspace. SinceH is normal inA 0 , this impliesA 0 leavesW invariant which is a contradiction. It follows by [Mc] that the only irreducible subgroups ofGL d (V ) for which a single orbit contains all 23 nonzero vectors in a hyperplane areSL d (V ) orSp d (V ), withd even in the last case. In the first case, A 0 = SL d (2) is 2–transitive on Vnf0g, so A is 3–transitive. Thus r 3 + 2 1 + 2< 4 n 2 =. In the second case,A 0;v has 3 orbits of nonzero vectors so, by Proposition 3.2.4,r 2 + 2 4 < 8 n 2 = , asn 16. Lemma 3.2.3 shows that in both casess 0 > 2=n. We summarize these results in the following theorem. Theorem 4.0.8. LetA andG be as above. Then one of the following holds: (a) r 2 = 0,s 0 = 0 and (A;G;V ) is exceptional; (b) r 2 = 1,s 0 = 1=n andA =G is Frobenius of ordern(n 1) withn =p d . (c) r 2 = 2,s 0 = 2=n andA =G is Frobenius of ordern(n 1)=2 withn =p d andp is odd. (d) s 0 > 2=n; (e) A =S 4 ,G =A 4 ands 0 = 2=4. 24 Chapter 5 Regular Normal Nonabelian Subgroup Lemma 5.0.9. LetA,G be transitive subgroups withA=G cyclic generated byxG. Sup- pose that for any xg2 xG, we know that either xg is a derangement or xg has at least s> 1 fixed points. Then the proportion of derangements inxG is at least 1 1=s 1=2. Proof. Letd denote the number of derangements inxG. By Lemma 3.1.3, since bothA andG act transitively onX, we have 1 = 1 jGj X xg2xG jFix(xg)j s(jGjd) jGj ; which, by solving ford=jGj, yields the required inequality. Remark 5.0.10. In the above inequality, equality holds precisely whens 0 =s 2 = 1=2. Lemma 5.0.11. Let A be a finite group acting transitively on a set X. If N is a normal regular subgroup of A, then if g 2 A has a fixed point, the number of fixed points is jC N (g)j. Proof. Let 2 Fix(g) be a fixed point of g. Let f : C N (g)! Fix(g) be defined by f(x) =x(). 1. x2 C N (g)) gx = xg) g(x()) = x(g()) = x()) x()2 Fix(g), so f is well-defined. 25 2. Letx;y2 C N (g) such thatf(x) = f(y), which meansx() = y(),y 1 x() = . SinceN is a regular subgroup,x =y and thusf is injective. 3. Let2 Fix(g). AsN is regular, and thus transitive, there exists a (unique) element x2N such thatx() =. Since;2Fix(g) we have x() =)xg() =g())g 1 xg() =: SinceN is normalg 1 xg2N and asN is regularg 1 xg() = =x() impliesg 1 xg = x, thus x2 C N (g) and therefore f is surjective. It follows that f is a bijection and so jC N (g)j =jFix(g)j. Proposition 5.0.12. Let N be a finite group and x2 Aut(N) with C N (x) = 1 (x fixes only the identity). ThenN =fg 1 g x :g2Ng. Proof. Letg;h2N so thath 1 h x =g 1 g x . Thenhg 1 = (hg 1 ) x and thereforehg 1 = 1) g = h. Thus g! g 1 g x is an injective map from a finite set to itself, therefore a bijection. Lemma 5.0.13. Let N be a finite group and x2 Aut(N) with C N (x) = 1. Then N is solvable. Proof. By contradiction, assume there exist nonsolvable groups verifying the hypothesis and pick such group G withjGj minimal. Let A denote the semidirect product Go < x >. Sinceh 1 xh = h 1 xhx 1 x = h 1 h x 1 x, by Proposition 5.0.11 (applied tox 1 ) it follows that the hypothesis is equivalent to Gx being a single conjugacy class in A. As the hypothesis holds forx-invarint sections ofG, one may assumeG is characteristically 26 simple. By 12.1 [FGS], there exists an involution t2 G with t G = t A which implies A = GC A (t). Then there exists g 2 G such that gx 2 C A (t) so t 2 C G (gx) which is therefore nontrivial. Asgx andx are conjugates (via an element ofG), it follows that C G (x) is nontrivial, a contradiction. Notice that this immediately implies: Corollary 5.0.14. IfN is a direct product of simple nonabelian groups andx2 Aut(N), thenjC N (x)j> 1. In the setting of Lemma 5.0.9, any elementg2A can be view as an element ofAut(N) as acting by conjugation. Moreover, the centralizer ofg inN is the same as the set of fixed points ofg acting by conjugation onN. AssumingA is primitive and not affine, the socle ofA isH'T m , withT simple nonabelian. IfH is regular (one of the cases deriving from the Aschbacher–O’Nan–Scott Theorem), then Theorem 5.0.15. Let A be a primitive permutation and G a normal subgroup such that A=G is cyclic. Assume that the socle ofA is regular nonabelian. Thens 0 1 2 . On the other hand, by Remark 3.2.6, if we assume that r 2 = 1, then it follows that s 0 =s 2 = 1=2. 27 References [BLP] J.P. Buhler, H. W. Lenstra and C. Pomerance, Factoring integers with the number field sieve, in The Development of The Number Field Sieve, Lecture Notes in Mathe- matics 1554, Springer-Verlag, Berlin, 1993. [BS] B. J. Birch and H. P. F. Swinnerton-Dyer, Note on a problem of Chowla, Acta Arith- metica 5 (1959), 417-423. [CC] P. J. Cameron and A. M. Cohen, On the number of fixed point free elements in a permutation group, Annals of Discrete Mathematics 106/107 (1992), 135-138. [Ch] S. Chowla, The Riemann zeta and allied functions, Bulletin of the American Mathe- matical Society 58 (1952), 287-303. [DM] J. D. Dixon and Brian Mortimer, Permutation Groups, Springer-Verlag New York- Berlin-Heidelberg, 1996 [FGS] M. Fried, R. Guralnick and J. Saxl, Schur covers and Carlitz’s conjecture, Israel Journal of Mathematics 82 (1993), 157-225. [GW] R. Guralnick and D. Wan, Bounds for Fixed Point Free Elements in a Transitive Group and Applications to Curves over Finite Fields, Israel Journal of Mathematics 101 (1997), 255-287. [Jo] C. Jordan, Recherches sur les substitutions, J. des Math. Pures et Appl. (Liouville) 17 (1872), 351-367 (Oeuvres, I, no. 52). [M] P.R. de Montmort, Essay d’analyse sur les jeux de hazard, (1708) 1st ed. , (1713) (2nd ed.). Jacques Quillau, Paris. Reprinted 1980 by Chelsea, New York. [Mc] J. McLaughlin, Some subgroups of Sl n (F 2 ), Illinois Journal of Mathematics 13 (1969), 108-115. [Wi] R. A. Wilson, The Finite Simple Groups, Springer London Dordrecht Heidelberg New York, 2009 28
Abstract (if available)
Abstract
Motivated by questions arising in connection with branched coverings of connected smooth projective curves over finite fields, I am studying the proportion of fixed point free elements (derangements) in cosets of normal subgroups of primitive permutations groups. Using the Aschbacher–O’Nan–Scott theorem for primitive groups to partition the problem, I provide complete answers for affine groups and groups which contain a regular normal nonabelian subgroup.
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Pavelescu, Andrei Bogdan
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On the proportion of derangements in cosets of primitive permutation groups
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College of Letters, Arts and Sciences
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Doctor of Philosophy
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Mathematics
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04/12/2012
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arithmetic monodromy,derangements,fixed point free elements,OAI-PMH Harvest,permutation groups,primitive
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