Attila Maróti

Covering the symmetric groups with proper subgroups

Let G be a group that is a set-theoretic union of finitely many proper subgroups. Cohn defined σ(G) to be the least integer m such that G is the union of m proper subgroups. Tomkinson showed that σ(G) can never be 7, and that it is always of the form q + 1 (q a prime power) for solvable groups G. In this paper we give exact or asymptotic formulas for σ(Sn). In particular, we show that σ(Sn) ≤ 2n-1, while for alternating groups we find σ(An) ≥ 2n-2 unless n = 7 or 9. An application of this result is also given.

Bounding the number of conjugacy classes of a permutation group

Abstract For a finite group G, let k(G) denote the number of conjugacy classes of G. If G is a finite permutation group of degree n > 2, then k(G) ≤ 3 (n −1)/2. This is an extension of a theorem of Kovács and Robinson and in turn of Riese and Schmid. If N is a normal subgroup of a completely reducible subgroup of GL(n, q), then k(N ) ≤ q5n . Similarly, if N is a normal subgroup of a primitive subgroup of Sn , then k (N ) ≤ p (n) where p (n) is the number of partitions of n. These bounds improve results of Liebeck and Pyber.

Ring elements as sums of units

In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph Γ(R) is a power of 2.

A Proof of a Generalized Nakayama Conjecture

In a recent paper Kulshammer, Olsson, and Robinson proved a deep generalization of the Nakayama conjecture for symmetric groups. We provide a similar but a shorter and relatively elementary proof of their result. Our method enables us to obtain a more general H-analogue of the Nakayama conjecture where H is a set of positive integers.

Symmetric functions, generalized blocks, and permutations with restricted cycle structure

We present various techniques to count proportions of permutations with restricted cycle structure in finite permutation groups. For example, we show how a generalized block theory for symmetric groups, developed by Kulshammer, Olsson, and Robinson, can be used for such calculations. The paper includes improvements of recurrence relations of Glasby, results on average numbers of fixed points in certain permutations, and a remark on a conjecture of Robinson related to the so-called k(GV)-problem of representation theory. We extend and give alternative proofs for previous results of Erdos and Turan; Glasby; and Beals, Leedham-Green, Niemeyer, Praeger and Seress.

Normal coverings of linear groups

For a non-cyclic nite group G, let (G) denote the smallest number of conjugacy classes of proper subgroups of G needed to cover G. Let (G) denote the size of the largest set of conjugacy classes of G, such that any two elements from distinct classes generate G. In this paper several explicit bounds or formulas are given for (G) and (G), where G is a subgroup of GLn(q) containing SLn(q). The results hold also for the group G=Z(G). Motivated by questions in number theory, Bubboloni and Praeger have recently given bounds or exact formulas for (Sn) and (An), for all values of n. Further work of Bubboloni, Praeger and Spiga has established that (Sn) is bounded above and below by linear functions of n. This paper establishes a similar result for linear groups: it is shown that n= 2 2; the upper bound is exact in the case that n is an odd prime.

Covering certain wreath products with proper subgroups

Abstract For a non-cyclic finite group X let σ(X) be the least number of proper subgroups of X whose union is X. Precise formulas or estimates are given for σ(S ≀ Cm ) for certain non-abelian finite simple groups S where Cm is a cyclic group of order m.

ON THE GENERATING GRAPH OF A SIMPLE GROUP

The authors were supported by Universita di Padova (Progetto di Ricerca di Ateneo: Invariable generation of groups). The second author was also supported by an Alexander von Humboldt Fellowship for Experienced Researchers, by OTKA grants K84233 and K115799, and by the MTA Renyi Lendulet Groups and Graphs Research Group.

A proof of Pyber's base size conjecture

Abstract Building on earlier papers of several authors, we establish that there exists a universal constant c > 0 such that the minimal base size b ( G ) of a primitive permutation group G of degree n satisfies log ⁡ | G | / log ⁡ n ≤ b ( G ) 45 ( log ⁡ | G | / log ⁡ n ) + c . This finishes the proof of Pybers base size conjecture. The main part of our paper is to prove this statement for affine permutation groups G = V ⋊ H where H ≤ G L ( V ) is an imprimitive linear group. An ingredient of the proof is that for the distinguishing number d ( G ) (in the sense of Albertson and Collins) of a transitive permutation group G of degree n > 1 we have the estimates | G | n d ( G ) ≤ 48 | G | n .

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

We consider factorizations of a finite group G into conjugate subgroups, G = Ax1⋯Axk for A ≤ G and x1,…,xk ∈ G, where A is nilpotent or solvable. We derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group which, for a large class of groups, is linear in the non-solvable length of G. We also show that every solvable group G is a product of at most 1 + clog |G : C| conjugates of a Carter subgroup C of G, where c is a positive real constant. Finally, using these results we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.