Famously, every finite simple group G can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate G tends to 1 as |G| →∞. In this paper, we generalize this theorem of Liebeck and Shalev. Work of Lucchini and Menegazzo (1997) implies that a finite group G can be generated by a pair of elements if it has a unique chief series. As a consequence of our main theorem, the probability that a pair of elements generate such a group G tends to 1 as |S| →∞, where S is the unique simple quotient of G. We also prove that a profinite group G with finitely many chief series has a topological generating set of size d < ∞, and for any such d, the probability that a d-tuple of elements topologically generates G is positive; moreover, we can take d = 2 if G has a unique chief series. Along the way, we show that the chief factors of a finite group with a unique chief series are highly constrained, and we also analyze the maximal subgroup zeta function of a finite group with a unique minimal normal subgroup.

We introduce the concept of a type system P \mathcal {P} , that is, a partition on the set of finite words over the alphabet { 0 , 1 } \{0,1\} compatible with the partial action of Thompson’s group V V , and associate a subgroup Stab V ⁡ ( P ) \operatorname {Stab}_{V}(\mathcal {P}) of V V . We classify the finite simple type systems and show that the stabilizers of various simple type systems, including all finite simple type systems, are maximal subgroups of V V . We also find an uncountable family of pairwise nonisomorphic maximal subgroups of V V . These maximal subgroups occur as stabilizers of infinite simple type systems and have not been described in previous literature: specifically, they do not arise as stabilizers in V V of finite sets of points in Cantor space. Finally, we show that two natural conditions on subgroups of V V (both related to primitivity) are each satisfied only by V V itself, giving new ways to recognise when a subgroup of V V is not actually proper.

We show that R. Thompson’s group T is a maximal subgroup of the group V. The argument provides examples of foundational calculations which arise when expressing elements of V as products of transpositions of basic clopen sets in the Cantor space ℭ .

Let G(1,) G(2),& mldr; be a sequence of groups each of which is either an alternating group, a symmetric group or a cyclic group. Let us construct a sequence (W-i) of wreath products via W-1 = G(1 )and, for each i >= 1, Wi+1 = G(i+1)wr W-i via the natural permutation action. We determine the minimum number d(W-i) of generators required for each wreath product in this sequence.

AbstractThe higher-dimensional Thompson groups $nV$ , for $n \geqslant 2$ , were introduced by Brin [‘Presentations of higher dimensional Thompson groups’, J. Algebra284 (2005), 520–558]. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in $nV$ and reflecting the self-similar structure of n-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.

Just infinite groups play a significant role in profinite group theory. For each $c \geq 0$, we consider more generally JNN$_c$F profinite (or, in places, discrete) groups that are Fitting-free; these are the groups $G$ such that every proper quotient of $G$ is virtually class-$c$ nilpotent whereas $G$ itself is not, and additionally $G$ does not have any non-trivial abelian normal subgroup. When $c = 1$, we obtain the just non-(virtually abelian) groups without non-trivial abelian normal subgroups. Our first result is that a finitely generated profinite group is virtually class\nbd$c$ nilpotent if and only if there are only finitely many subgroups arising as the lower central series terms $\gamma_{c+1}(K)$ of open normal subgroups $K$ of $G$. Based on this we prove several structure theorems. For instance, we characterize the JNN$_c$F profinite groups in terms of subgroups of the above form $\gamma_{c+1}(K)$. We also give a description of JNN$_c$F profinite groups as suitable inverse limits of virtually nilpotent profinite groups. Analogous results are established for the family of hereditarily JNN$_c$F groups and, for instance, we show that a Fitting-free JNN$_c$F profinite (or discrete) group is hereditarily JNN$_cF$ if and only if every maximal subgroup of finite index is JNN$_c$F. Finally, we give a construction of hereditarily JNN$_c$F groups, which uses as an input known families of hereditarily just infinite groups.

Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schutzenberger groups. The main result of this paper states that for a finitely generated commutative semigroup S, these three separability conditions coincide and are equivalent to every H-class of S being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely generated semigroups. For a semigroup with finitely many H-classes, we investigate whether it has one of these properties if and only if all its Schtitzenberger groups have the property.

We investigate four finiteness conditions related to residual finiteness: complete separability, strong subsemigroup separability, weak subsemigroup separability and monogenic subsemigroup separability. For each of these properties we examine under which conditions the property is preserved under direct products. We also consider if any of the properties are inherited by the factors in a direct product. We give necessary and sufficient conditions for finite semigroups to preserve the properties of strong subsemigroup separability and monogenic subsemigroup separability in a direct product.

We investigate the structure of the monoid of endomorphisms of the ordered set (Q, <=) of rational numbers. We show that for any countable linearly ordered set Omega, there are uncountably many maximal subgroups of End (Q, <=) isomorphic to the automorphism group of Omega. We characterize those subsets X of Q that arise as a retract in (Q, <=) in terms of topological information concerning X. Finally, we establish that a countable group arises as the automorphism group of a countable linearly ordered set, and hence as a maximal subgroup of End (Q, <=), if and only if it is free abelian of finite rank.

We show that one can naturally describe elements of R. Thompson's finitely presented in finite simple group V, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions. This perspective provides an intuitive explanation towards the simplicity of V and also perhaps indicates a reason as to why it was one of the first discovered in finite finitely presented simple groups: it is (in some basic sense) a relative of the finite alternating groups. We find a natural in finite presentation for V as a group generated by these "transpositions," which presentation bears comparison with Dehornoy's in finite presentation and which enables us to develop two small presentations for V : a human-interpretable presentation with three generators and eight relations, and a Tietze-derived presentation with two generators and seven relations.

Abstract We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph R. As a consequence we show that, for any countable graph Γ, there are uncountably many maximal subgroups of the endomorphism monoid of R isomorphic to the automorphism group of Γ. Further structural information about End R is established including that Aut Γ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.

We study the probability of generating a finite simple group, together with its generalisation P G,socG (d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/socG. We prove that P G,socG (2) ⩽ 53/90, with equality if and only if G is A6 or S6, and establish a similar result for P G,socG (3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel’nikov on profinite groups, follow easily from our results.

For a semigroup S its d-sequence is d(S)=(d 1,d 2,d 3,…), where d i is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.

Introduction C. M. Campbell and E. F. Robertson 1. Algorithms for matrix groups E. A. O'Brien 2. Residual properties of 1-relator groups Mark Sapir 3. Words and groups Dan Segal 4. The modular isomorphism problem for the groups of order 512 Bettina Eick and Alexander Konovalov 5. Recent progress in the symmetric generation of groups Ben Fairbairn 6. Discriminating groups: a comprehensive overview Benjamin Fine, Anthony M. Gaglione, Alexei Myasnikov, Gerhard Rosenberger and Dennis Spellman 7. Extending the Kegel-Wielandt theorem through pi-decomposable groups L. S. Kazarin, A. Martinez-Pastor and M. D. Perez-Ramos 8. On the prime graph of a finite group Behrooz Khosravi 9. Applications of Lie rings with finite cyclic grading E. I. Khukhro 10. Pronormal subgroups and transitivity of some subgroup properties Leonid A. Kurdachenko, Javier Otal and Igor Ya. Subbotin 11. On Engel and positive laws O. Macedonska and W. Tomaszewski 12. Maximal subgroups of odd index in finite groups with simple classical socle N. V. Maslova 13. Some classic and nearly classic problems on varieties of groups Peter M. Neumann 14. Generalizations of the Sylow theorem Danila O. Revin and Evgeny P. Vdovin 15. Engel groups Gunnar Traustason 16. Lie methods in Engel groups Michael Vaughan-Lee 17. On the degree of commutativity of p-groups of maximal class A. Vera-Lopez and M. A. Garcia-Sanchez 18. Class preserving automorphisms of finite p-groups: a survey Manoj K. Yadav 19. Symmetric colorings of finite groups Yuliya Zelenyuk.

We study finite groups for which every minimal generating set has the same cardinality. A group has the basis property if it and every subgroup satisfies this condition on minimal generating sets. We classify all finite groups with the basis property.

In this article we use the Classification of the Finite Simple Groups, the O'Nan–Scott Theorem, and Aschbacher's theorem to classify the primitive permutation groups of degree less than 4096. The results will be added to the primitive groups databases of GAP and MAGMA.

Let Omega(Omega) be the semigroup of all mappings of a countably infinite set Omega. If U and V are subsemigroups of Omega(Omega), then we write U approximate to V if there exists a finite subset F of Omega(Omega) such that the subsemigroup generated by U and F equals that generated by V and F. The relative rank of U in Omega(Omega). is the least cardinality of a subset A of Omega(Omega) such that the union of U and A generates Omega(Omega). In this paper we study the notions of relative rank and the equivalence approximate to for semigroups of endomorphisms of binary relations on Omega.The semigroups of endomorphisms of preorders, bipartite graphs, and tolerances on Omega are shown to lie in two equivalence classes under approximate to. Moreover such semigroups have relative rank 0, 1, 2, or partial derivative in Omega(Omega) where partial derivative is the minimum cardinality of a dominating family for N(N). We give examples of preorders, bipartite graphs, and tolerances on Omega where the relative ranks of their endomorphism semigroups in Omega(Omega) are 0, 1, 2, and partial derivative.We show that the endomorphism semigroups of graphs, in general, fall into at least four classes under approximate to and that there exist graphs where the relative rank of the endomorphism semigroup is 2(N0). (C) 2010 Elsevier By. All rights reserved.

Abstract For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An 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.