Martyn Quick

Probabilistic Generation of Wreath Products of Non-abelian Finite Simple Groups

Abstract We show that the probability of generating an iterated standard wreath product of non-abelian finite simple groups is positive and tends to 1 as the order of the first simple group tends to infinity. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated. Information depending on the Classification of Finite Simple Groups is used throughout.

PROBABILISTIC GENERATION OF WREATH PRODUCTS OF NON-ABELIAN FINITE SIMPLE GROUPS, II

We show that the probability of generating an iterated wreath product of non-abelian finite simple groups converges to 1 as the order of the first simple group tends to infinity provided the wreath products are constructed with transitive and faithful actions. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated.

GROWTH OF GENERATING SETS FOR DIRECT POWERS OF CLASSICAL ALGEBRAIC STRUCTURES

For an algebraic structure A denote by d.A/ the smallest size of a generating set for A, and let d.A/D.d.A/; d.A 2 /; d.A 3 /; : : :/, where A n denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d.A/ when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d.A/ grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d.A/ is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.

Generating transformation semigroups using endomorphisms of preorders, graphs, and tolerances

Abstract Let Ω Ω be the semigroup of all mappings of a countably infinite set Ω . If U and V are subsemigroups of Ω Ω , then we write U ≈ V if there exists a finite subset F of Ω Ω such that the subsemigroup generated by U and F equals that generated by V and F . The relative rank of U in Ω Ω is the least cardinality of a subset A of Ω Ω such that the union of U and A generates Ω Ω . In this paper we study the notions of relative rank and the equivalence ≈ for semigroups of endomorphisms of binary relations on Ω . The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Ω are shown to lie in two equivalence classes under ≈ . Moreover such semigroups have relative rank 0 , 1 , 2 , or d in Ω Ω where d is the minimum cardinality of a dominating family for N N . We give examples of preorders, bipartite graphs, and tolerances on Ω where the relative ranks of their endomorphism semigroups in Ω Ω are 0 , 1 , 2 , and d . We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under ≈ and that there exist graphs where the relative rank of the endomorphism semigroup is 2 ℵ 0 .

Maximal complements in wreath products

ThenX wr∆ Y is the semidirect product of K by Y via this action. We call K thebase group of the wreath product. There is a natural action of X wr∆ Y on the Cartesian product Γ ×∆. For full details on the construction of the wreath product, see Section 1.6 of Robinson Our goal in this paper is to classify the maximal subgroups of the wreath product w complement the base group. This work was originally motivated by the possibili obtaining a simultaneous extension of the second author’s work on probabilistic gene of iterated wreath products of non-abelian simple groups (see [9]) and Bhattacharje result concerning probabilistic generation of iterated wreath products of alternating g This extension obtained by applying our work will appear in [10]. Bhattacharjee cons the sequence of groups defined as follows. Let m1,m2, . . . be a sequence of positiv

Maximal Complements in Finite Groups

Let G be a finite group with a non-Abelian minimal normal subgroup N which is a direct product of copies of the simple group X. A parametrization is given for the conjugacy classes of maximal subgroups of G which complement N in terms of certain homomorphisms taking values in Aut X.