Derangements and p -elements in permutation groups Peter J. Cameron p.j.cameron@qmul.ac.uk Groups and their ApplicationsManchester, 14 February 2007 uous closed curve in S which cannot be lifted In the beginning . . . A derangement is a permutation with no fixed to T . points. • The Fein–Kantor–Schacher theorem (see later) is equivalent to the statement that the relative 1. The proportion of derangements in the sym- Brauer group of any finite extension of global metric group S n is approximately 1/e. fields is infinite. (The proof uses the classifi- More precisely, the number of derangements in cation of finite simple groups.) S n is the nearest integer to n !/e. 2. (Jordan) A transitive permutation group of So find one then . . . degree n > 1 contains a derangement. A subgroup of S n can be generated by at most n − 1 elements, and such a generating set can In fact (Cameron and Cohen) the proportion of derangements in a transitive group G is at least be found efficiently (with polynomial delay) (Jer- 1/ n . rum). So such a subgroup can be described by O ( n 2 log n ) bits. Equality holds if and only if G is sharply 2 - transitive , and hence is the affine group { x �→ ax + b : a , b ∈ F , a � = 0 } over a nearfield F . Problem: Given a subgroup of S n , does it con- tain a derangement? The finite nearfields were determined by Zassenhaus. They all have prime power order. This problem is NP-complete, even for elemen- tary abelian 2-groups. There is a simple reduction Why do we care? from the known NP-complete problem 3-SAT. In- The presence of derangements in a permutation deed, the argument shows that counting the de- group has important implications in number the- rangements in a subgroup of S n is #P-complete. ory and topology. See Serre’s beautiful paper “On a theorem of Jordan”, in Bull. Amer. Math. Soc. 40 and in a transitive group . . . (2003), 429–440. Given generators for a subgroup G of S n , we can • Let f be an integer polynomial of degree n > check quickly whether H is transitive. If it is (and 1, irreducible over Q . Then f has no roots n > 1), then we know that G contains a derange- mod p for infinitely many primes p (indeed, ment. for at least a proportion 1/ n of all primes). Problem: Suppose that G is transitive. Find a derangement in G . • Let π : T → S be a covering map of de- gree n ≥ 2, and suppose that T is arcwise con- There is an efficient randomised algorithm for nected but not empty. Then there is a contin- this problem. Since at least a fraction 1/ n of the 1
elements of G are derangements, we can do this by The proof uses the Classification of Finite Sim- random search: in n trials we will have a better- ple Groups, together with detailed analysis of the than-even chance of finding one, and in n 2 trials various families of simple groups. we will fail with exponentially small probability. Problem: Find a simple proof! Problem: Can it be done deterministically? The answer is likely to be “yes” – this is theo- retically interesting but the randomised algorithm Prime order will almost certainly be more efficient! Not every transitive group contains a derange- ment of prime order. A simple example is the 1-dimensional affine Groups with many derangements group over GF ( 9 ) , acting on the set of 12 lines of Although the lower bound | G | / n for the num- the affine plane of order 3. ber of derangements in a transitive group G is at- tained (by sharply 2-transitive groups), there are Call a transitive group elusive if it contains no many groups with a higher proportion of derange- derangement of prime order. ments. For example, if G is regular, than all but one of its elements are derangements! Problem: Is it true that the degrees of elusive groups have density zero? The argument of Cameron and Cohen gives a lower bound of about ( r − 1 ) / n for the proportion A permutation group G is 2 -closed if any permu- of derangements in a transitive group G , where r tation which fixes every G -orbit on 2-sets belongs is the permutation rank (the number of orbits of G to G . For example, the automorphism group of a on ordered pairs). graph is 2-closed. Can anything be said about families of (say, primitive) groups in which the proportion of de- Problem: Is it true that there is no 2-closed elu- rangements is bounded away from zero? sive group? (Klin) An example Which prime? Example: There is a constant α k > 0 so that the Conjecture: For any prime p , there is a func- proportion of derangements in S n acting on k -sets tends to α k as k → ∞ . (For example, α 1 = e − 1 = tion f p on the natural numbers such that, if G 0.3679 . . . , while α 2 = 2e − 3/2 = 0.4463 . . . . is a transitive group of degree n = p a b , where gcd ( p , b ) = 1 and a ≥ f p ( b ) , then G contains a There is a formula for α k as a sum over subsets derangement of p -power order. of the partitions of k . But most of the terms cancel, This was conjectured for p = 2 by Isbell in 1959 so I suspect there is a much simpler formula! (in the context of game theory); even that case is Problem: Is it true that α k → 1 monotonically as still open. k → ∞ ? Conjecture: For any prime p , there is a func- tion g p on the natural numbers such that, if P is Prime power order Theorem: A transitive group of degree n > 1 a p -group with b orbits each of length greater than g p ( b ) , then P contains a derangement. contains a derangement. (Jordan) The second conjecture implies the first. The proof is elementary: By the Orbit-counting Lemma, the average number of fixed points is 1; Maillet and Blichfeldt and some element (the identity) fixes more than one point. Theorem 1 (Maillet 1895) . Let G be a permutation group of degree n, and L the set of numbers of fixed Theorem: A transitive group of degree n > points of non-trivial subgroups of G. Then | G | divides 1 contains a derangement of prime-power order. ∏ l ∈ L ( n − l ) . (Fein–Kantor–Schacher) 2
Theorem 2 (Blichfeldt 1904) . Let G be a permutation (wreath products of regular groups with symmet- ric groups; alternating groups; extensions of V r by group of degree n, and L the set of numbers of fixed points of non-identity elements of G. Then | G | divides G , where G is the stabiliser of a tuple of points in a ∏ l ∈ L ( n − l ) . general linear group and V its natural module. In addition there are some sporadic examples with Blichfeldt claimed that his was just a new proof | L | small. of Maillet’s Theorem but it is actually stronger. The classification of groups meeting Blichfeldt’s bound is not known. Proof of Maillet’s Theorem The proof is by induction on n . Let l 0 = min ( L ) . Prime power versions If l 0 � = 0, then G fixes l 0 points; removing these In both Maillet’s and Blichfeldt’s Theorems (as fixed points subtracts l 0 from n and from every el- they both observed), we can take a smaller set L , ement of L and leaves ∏ l ∈ L ( n − l ) unaltered. the set of fixed point numbers of non-trivial ele- If 0 ∈ L , then any point stabiliser satisfies the ments or subgroups of prime-power order. hypotheses with L replaced by L \ { 0 } . By in- For, with this hypothesis, each Sylow subgroup duction, the order of any point stabiliser divides satisfies the divisibility condition, and so the ∏ l ∈ L \{ 0 } ( n − l ) . Since n divides the least common whole group does. multiple of the orbit lengths, it follows that | G | di- Problem 3. Which groups attain the bounds in the vides ∏ l ∈ L ( n − l ) . prime power versions of Maillet’s or Blichfeldt’s The- orems? Equality implies that the pointwise stabiliser of any set is transitive on the points it moves. Such a Partitions group acts on a nice geometry (a matroid, indeed A partition of a finite group G is a set of non- a “perfect matroid design”, that is, a matroid in identity proper subgroups such that every non- which the cardinality of a flat depends only on its identity element is contained in exactly one of dimension. these subgroups. Iwahori and Kondo showed in 1965 that a group Proof of Blichfeldt’s Theorem G has a partition if and only if it has a permuta- The function tion representation in which every non-identity el- ement has k fixed points, for some k > 0 (the case g �→ ∏ ( fix ( g ) − l ) | L | = 1 in Blichfeldt’s Theorem). l ∈ L Suzuki showed in 1961 that a non-solvable group having a partition is one of PGL ( 2, q ) , is a virtual character which is zero on all non- PSL ( 2, q ) or Sz ( q ) for some prime power q . identity elements. So it is a multiple of the regu- lar character, whence | G | divides its value at the Local partitions identity. Analogous results hold for prime power ele- It is not at all clear what the consequence of ments. A A local partition of a group is a set of equality is, apart from saying that the above func- non-identity proper subgroups so that each non- tion is the regular character of G . identity local element (element of prime power or- der) is contained in exactly one of them. Which groups meet the bound? Spiga showed that a group has a local partition Any regular permutation group attains both if and only if it has a permutation representation in bounds, with L = { 0 } . which each non-identity local element has k fixed Groups meeting the bound in Maillet’s Theorem points, for some k > 0. have been determined by Zil’ber for | L | ≥ 7 us- He also found the finite simple groups which ing geometric and model-theoretic methods, and have local partitions: in addition to those in by Maund for | L | ≥ 2 using the Classification of Suzuki’s list, the Ree groups R 1 ( q ) and the first Finite Simple Groups. There are generic examples Janko group J 1 are the only ones. 3
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