László Pyber

On Random Generation of the Symmetric Group

We prove that the probability i ( n, k ) that a random permutation of an n element set has an invariant subset of precisely k elements decreases as a power of k , for k ≤ n /2. Using this fact, we prove that the fraction of elements of S n belong to transitive subgroups other than S n or A n tends to 0 when n → ∞, as conjectured by Cameron. Finally, we show that for every ∈ > 0 there exists a constant C such that C elements of the symmetric group S n , chosen randomly and independently, generate invariably S n with probability at least 1 − ∈. This confirms a conjecture of McKay.

A new generalization of the Erdo¨s-Ko-Rado theorem

Abstract Let A and B be systems of k and l element subsets of an n element set respectively. Suppose that A ∩ B ≠ ⊘ for all A ϵ A , B ϵ B . It is proved that | A | | B ⩽ n −1 k −1 n −1 l −1 , whenever n ⩾ 2k + l − 2 (k ⩾ l).

Growth in finite simple groups of Lie type

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple groups L of Lie type of bounded rank the diameter of any Cayley graph is polylogarithmic in |L|. We obtain a similar bound for the diameters of all Cayley graphs of perfect subgroups of GL(n,p) generated by their elements of order p. We also obtain some new families of expanders. We also prove the following partial extension. Let G be a subgroup of GL(n,p), p a prime, and S a symmetric set of generators of G satisfying |S^3|\le K|S| for some K. Then G has two normal subgroups H\ge P such that H/P is soluble, P is contained in S^6 and S is covered by K^c cosets of H where c depends on n. We obtain results of similar flavour for sets generating infinite subgroups of GL(n,F), F an arbitrary field.

Covering a graph by complete bipartite graphs

We prove the following theorem: the edge set of every graph G on n vertices can be partitioned into the disjoint union of complete bipartite graphs such that each vertex is contained by at most c(n/log n) of the bipartite graphs.

On the orders of doubly transitive permutation groups, elementary estimates

Abstract Extending ideas of L. Babai we give an n c log 2 n bound for the orders of 2-transitive groups of degree n not containing A n . Our proof is elementary in the sense that it does not invoke the classification theorem of finite simple groups. We also give ideas leading to a short proof of a slightly weaker bound.

Maximal subgroups in finite and profinite groups

We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 203-220, we then prove that a finite group G has at most IGIc maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.

How to find many counterfeit coins

AbstractWe propose an algorithm for findingm defective coins, that uses at most

Regular subgraphs of dense graphs

\left\lceil {\log _3 \left( {\begin{array}{*{20}c} n \\ m \\ \end{array} } \right)} \right\rceil