We classify semigroups in the title according to whether they have a finite or an infinite number ofℒ-classes or ℛ-classes. For each case, we provide a concrete construction using Rees matrix semigroups and their translational hulls. An appropriate relatively free semigroup is used to complete the classification. All this is achieved by first treating the special case in which one of the generators is idempotent. We conclude by a discussion of a possible classification of 2-generator completely regular semigroups.

Let G be a finite group, let p be a prime divisor of the order of G and let k(G) be the number of conjugacy classes of G. By disregarding at most finitely many non-solvable p-solvable groups G, we have with equality if and only if if is an integer, and CG(Cp) = Cp. This extends earlier work of Héthelyi, Külshammer, Malle and Keller.

Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Yg for some g ∊ 〈X, Y〉, for all X ≤ A and Y ≤ B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups.

In this note, we prove that the Gauss–Picard modular group PU(2,1;Θ1) has Property (FA). Our result gives a positive answer to a question by Stover [‘Property (FA) and lattices in SU(2,1)’, Internat. J. Algebra Comput.17 (2007), 1335–1347] for the group PU(2,1;Θ1).

It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilizers. However, beside a well-known family of exceptional graphs, related to the lexicographic product of a cycle with an edgeless graph on two vertices, only a few such infinite families of graphs are known. In this paper, we present two more families of tetravalent arc-transitive graphs with large vertex-stabilizers, each significant for its own reason.

We describe a practical algorithm for primitivity testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For an imprimitive group, a system of imprimitivity can be constructed. An implementation of the algorithm in Magma is publicly available.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let 𝔤 be its Lie algebra. Let k(G), respectively, k(𝔤), be the field of k-rational functions on G, respectively, 𝔤. The conjugation action of G on itself induces the adjoint action of G on 𝔤. We investigate the question whether or not the field extensions k(G)/k(G)G and k(𝔤)/k(𝔤)G are purely transcendental. We show that the answer is the same for k(G)/k(G)G and k(𝔤)/k(𝔤)G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type An or Cn, and negative for groups of other types, except possibly G2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

The computation of growth series for the higher Baumslag–Solitar groups is an open problem first posed by de la Harpe and Grigorchuk. We study the growth of the horocyclic subgroup as the key to the overall growth of these Baumslag–Solitar groups BS(p,q), where 1<p<q. In fact, the overall growth series can be represented as a modified convolution product with one of the factors being based on the series for the horocyclic subgroup. We exhibit two distinct algorithms that compute the growth of the horocyclic subgroup and discuss the time and space complexity of these algorithms. We show that when p divides q, the horocyclic subgroup has a geodesic combing whose words form a context-free (in fact, one-counter) language. A theorem of Chomsky–Schützenberger allows us to compute the growth series for this subgroup, which is rational. When p does not divide q, we show that no geodesic combing for the horocyclic subgroup forms a context-free language, although there is a context-sensitive geodesic combing. We exhibit a specific linearly bounded Turing machine that accepts this language (with quadratic time complexity) in the case of BS(2,3) and outline the Turing machine construction in the general case.

We consider orientable closed connected 3-manifolds obtained by performing Dehn surgery on the components of some classical links such as Borromean rings and twisted Whitehead links. We find geometric presentations of their fundamental groups and describe many of them as 2-fold branched coverings of the 3-sphere. Finally, we obtain some topological applications on the manifolds given by exceptional surgeries on hyperbolic 2-bridge knots.

A conjecture of Gromov states that a one-ended word-hyperbolic group must contain a subgroup that is isomorphic to the fundamental group of a closed hyperbolic surface. Recent papers by Gordon and Wilton and by Kim and Wilton give sufficient conditions for hyperbolic surface groups to be embedded in a hyperbolic Baumslag double G. Using Nielsen cancellation methods based on techniques from previous work by the second author, we prove that a hyperbolic orientable surface group of genus 2 is embedded in a hyperbolic Baumslag double if and only if the amalgamated word W is a commutator: that is, W = [U, V] for some elements U, V ∈ F. Furthermore, a hyperbolic Baumslag double G contains a non-orientable surface group of genus 4 if and only if W = X2Y2 for some X, Y ∈ F. G can contain no non-orientable surface group of smaller genus.

Let Γ be a graph and let G be a vertex-transitive subgroup of the full automorphism group Aut(Γ) of Γ. The graph Γ is called G-normal if G is normal in Aut(Γ). In particular, a Cayley graph Cay(G, S) on a group G with respect to S is normal if the Cayley graph is R(G)-normal, where R(G) is the right regular representation of G. Let T be a non-abelian simple group and let G = Tℓ with ℓ ≥ 1. We prove that if every connected T-vertex-transitive cubic symmetric graph is T-normal, then every connected G-vertex-transitive cubic symmetric graph is G-normal. This result, among others, implies that a connected cubic symmetric Cayley graph on G is normal except for T ≅ A47 and a connected cubic G-symmetric graph is G-normal except for T ≅ A7, A15 or PSL(4, 2).

Let A be a finite group acting fixed-point freely on a finite (solvable) group G. A longstanding conjecture is that if (|G|, |A|) = 1, then the Fitting length of G is bounded by the length of the longest chain of subgroups of A. It is expected that the conjecture is true when the coprimeness condition is replaced by the assumption that A is nilpotent. We establish the conjecture without the coprimeness condition in the case where A is an abelian group whose order is a product of three odd primes and where the Sylow 2-subgroups of G are abelian.

Let Y be a fixed nonempty subset of a set X and let T(X,Y ) denote the semigroup of all total transformations from X into Y. In 1975, Symons described the automorphisms of T(X,Y ). Three decades later, Nenthein, Youngkhong and Kemprasit determined its regular elements, and more recently Sanwong, Singha and Sullivan characterized all maximal and minimal congruences on T(X,Y ). In 2008, Sanwong and Sommanee determined the largest regular subsemigroup of T(X,Y ) when |Y |≠1and Y ≠ X; and using this, they described the Green’s relations on T(X,Y ) . Here, we use their work to describe the ideal structure of T(X,Y ) . We also correct the proof of the corresponding result for a linear analogue of T(X,Y ) .

The partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.

Shumyatsky and the second author proved that if G is a finitely generated residually finite p-group satisfying a law, then, for almost all primes p, the fact that a normal and commutator-closed set of generators satisfies a positive law implies that the whole of G also satisfies a (possibly different) positive law. In this paper, we construct a counterexample showing that the hypothesis of finite generation of the group G cannot be dispensed with.

Erdős and Szekeres [‘Some number theoretic problems on binomial coefficients’, Aust. Math. Soc. Gaz.5 (1978), 97–99] showed that for any four positive integers satisfying m1+m2=n1+n2, the two binomial coefficients (m1+m2)!/m1!m2! and (n1+n2)!/n1!n2! have a common divisor greater than 1. The analogous statement for k-element families of k-nomial coefficients (k>1) was conjectured in 1997 by David Wasserman.

Erdős and Szekeres remark that if m1,m2,n1,n2 as above are all greater than 1, there is probably a lower bound on the common divisor in question which goes to infinity as a function of m1 +m2 .Such a bound is obtained in Section 2.

The remainder of this paper is devoted to proving results that narrow the class of possible counterexamples to Wasserman’s conjecture.

We count the number of lifts of an irreducible π-partial character that lies in a block with a cyclic defect group.

We prove that the variety consisting of all involutory inflations of normal bands is the unique maximal residually finite variety consisting of combinatorial semigroups with involution.

In this paper, we classify the finite groups belonging to the class of cyclic-transitive groups. A group G is said to be cyclic-transitive if the following condition holds: if x, y, z are nonidentity elements of G such that 〈x,y〉 and 〈y,z〉 are both cyclic, then 〈x,z〉 is also cyclic.

The topological complexity is a numerical homotopy invariant of a topological space X which is motivated by robotics and is similar in spirit to the classical Lusternik–Schnirelmann category of X. Given a mechanical system with configuration space X, the invariant measures the complexity of motion planning algorithms which can be designed for the system. In this paper, we compute the topological complexity of the configuration space of n distinct ordered points on an orientable surface, for both closed and punctured surfaces. Our main tool is a theorem of B. Totaro describing the cohomology of configuration spaces of algebraic varieties. For configuration spaces of punctured surfaces, this is used in conjunction with techniques from the theory of mixed Hodge structures.