s II, Acta Math. Acad. Sci. Hungar. 18 (1967), s S 151164; III, - PDF document

✁ ☞ ✂ ✝ � ✟ ✟ ✟ ✂ ✆ ✁ ✡ ✆ ☞ ✆ ✍ ✎ ✎ ✂ ✆ ✟ ✆ ✂ ✆ ✁ Extremal problems Permutations How many permutations in a set (or group) with prescribed distances? Peter J. Cameron School of Mathematical Sciences The distance between permutations g � h S n is the Queen Mary and Westfield College number of positions where g and h disagree (this is ✄ g London E1 4NS n ☎ 1 h ✆ ). fix U.K. ✄ n � n For S ✠ , let f S ✞ 0 2 ✆ be the size of the ✄ g Paul Erd˝ os Memorial Conference largest subset X of S n with fix ☎ 1 h S for all distinct ✄ n Budapest, Hungary � h g X ; for s n , let f s ✆ be the size of the largest ✄ n ✄ n 5 July 1999 s -distance subset of S n . Let f g ✆ and f g ✆ be the s S corresponding numbers for subgroups of S n . 1 3 Results and problems ✄ c 1 n ✄ n ✄ c 2 n ✆ 2 s ✆ 2 s . Erd˝ os and Tur´ ☛ s f s ☛ s an on random Theorem permutations ✄ f s ✄ n ✌ 2 s Problem Does s cn as n ∞ ? (for fixed ✆ 1 s , or for s ∞ ). . Erd˝ . Tur´ P os and P an, On some problems of a statistical group theory, ✄ n Theorem (Blichfeldt) f g I, Z. Wahrscheinlichkeitstheorie und Verw. Gebeite 4 ✆ divides S ✄ n (1965), 175–186; ∏ s II, Acta Math. Acad. Sci. Hungar. 18 (1967), s ✏ S 151–164; III, ibid. 18 (1967), 309–320; Problem Which groups attain Blichfeldt’s bound? IV, ibid. 19 (1968), 413–435; V, Period. Math. Hungar. 1 (1971), 5–13; VI, J. Indian Math. Soc. (N.S.) 34 (1971), 175–192; Problem Is it true that ☞ ∏ VII, Period. Math. Hungar. 2 (1972), 149–163. ✄ n ✄ n f S s s ✏ S for S fixed, n large? 2 4

✟ ☞ ✔ ✑ � ✟ ✟ ✟ ✂ ✠ ✆ ✖ ✆ ✟ ✆ ✑ � ✟ ✟ ✟ ✂ ✑ ✗ ✆ ☞ ✂ ✗ ✟ ✟ ✓ ✂ ✑ ✂ � ✟ ✟ ✟ ✒ ✂ ✒ ✒ ✆ ✆ ☞ A specific problem Derangements and Latin squares, � t Theorem (Blake–Cohen–Deza) If S � 1 ✞ 0 1 ✠ , continued then ✄ n ✄ n ✄ n f S n t 1 1 Problem Choose a random permutation π as Equality holds if and only if a sharply t -transitive set follows: select a Latin square from the uniform of permutations exists. distribution, normalise, and let π be the second row. (So the permutations which occur with positive � n Theorem If S ✕ S then ✞ 0 1 probability are the derangements.) ✄ n ✄ n f S ✒ f S n ! How does the ratio of the probability of the most and least likely derangement behave? � n Problem If S ✞ t ✠ , is 1 ✄ n ✄ n f S t ✆ ! for n large relative to t ? (The extremal configuration should be a coset of the Is it true that, with probability tending to 1 , a stabiliser of t points.) random derangement lies in no transitive subgroup of S n except S n and possibly A n ? The bound holds if a sharply t -transitive set exists. Compare the Erd˝ os–Ko–Rado theorem. 5 7 Derangements and Latin squares Derangements of prime power order A derangement is a permutation which has no fixed points. It is well-known that the number of Theorem (Frobenius) A non-trivial finite transitive derangements in S n is the nearest integer to n ! ☛ e . permutation group contains a derangement. If a Latin square of order n is normalised so that the ✄ 12 ✟ n Theorem (Kantor [CFSG]) A non-trivial finite first row is ✆ , then the other rows are transitive permutation group contains a derangement derangements. of prime power order. Every derangement occurs as the second row of a Problem (Isbell) Is it true that, if a is sufficiently large normalised Latin square. in terms of p and b ( p prime), then a transitive p a permutation group of degree n ✒ b contains a Problem Is it true that the distribution of the number derangement of p -power order? of rows of a random Latin square which are even ✄ n permutations is approximately binomial B � 1 ✆ ? 2 6 8

✟ � ✟ ✟ ✗ ✗ ✟ ✟ ✟ � ✠ Derangements of prime order Counting orbits Call G elusive if it is transitive and contains no The orbit-counting lemma asserts that the number of derangement of prime order. orbits of a finite permutation group G is equal to the average number of fixed points of elements of G . It is proved by counting edges in the bipartite graph on Theorem (Giudici [CFSG]) A quasiprimitive elusive � n ✙ G , where i is joined to g if g fixes i . group is isomorphic to M 11 ✘ H for some transitive ✞ 1 group H . � n Jerrum’s Markov chain on ✞ 1 ✠ : one step consists of two steps in a random walk on the graph. Problem Does the set of degrees of elusive groups The limiting distribution is uniform on the orbits. This have density zero? (This set contains 2 n for every gives a method for choosing random ‘unlabelled’ even perfect number n , and is multiplicatively closed.) structures. Problem (Jordan, Maruˇ siˇ c) Show that the Problem For which families of permutation groups is automorphism group of a vertex-transitive graph is this Markov chain rapidly mixing? non-elusive. 9 11 An infinite analogue Bertrand, Sylvester and Erd˝ os There is no natural way to choose a random Bertrand’s Postulate was proposed for an permutation of a countable set, since the symmetric application to permutation groups. The first published group is not compact. paper of Paul Erd˝ os was a short proof of Bertrand’s Postulate. Parallels: Sylvester generalised Bertrand’s Postulate as follows: The countable random graph (the generic Theorem The product of k consecutive numbers countable graph), Erd˝ os and R´ enyi. greater than k is divisible by a prime greater than k . A permutation of a finite set is given by a pair of Erd˝ os also gave a short proof of this. It deals with a total orders of the set. case in the proof of Giudici’s Theorem which cannot be handled by group-theoretic methods, where G is a symmetric or alternating group in its action on So instead of the random permutation, consider the k -element subsets. Sylvester’s Theorem gives a generic pair (or n -tuple) of total orders. Note that the derangement of prime order in this case. generic (or random) total order is isomorphic to Q . 10 12