We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms which we term F-morphisms. These play the same role in the theory of idempotent-pure homomorphisms that F-inverse monoids play in the theory of E-unitary inverse semigroups.

Various lattices of subgroups of a finite transitive permutation group G can be used to define a set of ‘basic’ permutation groups associated with G that are analogues of composition factors for abstract finite groups. In particular G can be embedded in an iterated wreath product of a chain of its associated basic permutation groups. The basic permutation groups corresponding to the lattice L of all subgroups of G containing a given point stabiliser are a set of primitive permutation groups. We introduce two new subgroup lattices contained in L, called the seminormal subgroup lattice and the subnormal subgroup lattice. For these lattices the basic permutation groups are quasiprimitive and innately transitive groups, respectively.

The aim of this work is to offer a new characterization of the Hilbert symbol Q*p from the commutator of a certain central extension of groups. We obtain a characterization for Q*p (p≠2) and a different one for Q*2.

Hypercentrally embedded subgroups of finite groups can be characterized in terms of permutability as those subgroups which permute with all pronormal subgroups of the group. Despite that, in general, hypercentrally embedded subgroups do not permute with the intersection of pronormal subgroups, in this paper we prove that they permute with certain relevant types of subgroups which can be described as intersections of pronormal subgroups. We prove that hypercentrally embedded subgroups permute with subgroups of prefrattini type, which are intersections of maximal subgroups, and with F-normalizers, for a saturated formation F. In the soluble universe, F-normalizers can be described as intersection of some pronormal subgroups of the group.

We define a notion of conjugacy in singular Artin moniods, and solve the corresponding conjugacy problem for finite types. We sgiw that this definition is appropriate to describe type (1) singular Markov moves on singular braids. Parabolic submonoids of singular Artin monoids are defined and, in finite type, are shown to be singular Artin monoids. Solutions to conjugacy-type problems of parabolic submonoids are described. Geometric objects defined by Fenn, Rolfsen and Zhu, called (j, k)-bands, are algebraically characterised, and a procedure is given which determines when a word represents a (j, k)-band.

Suppose that the finite group G acts faithfully and irreducibly on the finite G-module V of characteristic p not dividing |G|. The well-known k(GV)-problem states that in this situation, if k(G V) is the number of conjugacy classes of the semidirect product GV, then k(GV) ≤ |V|. For p—solvable groups, this is equivalent to Brauer's famous k(B)-problem. In 1996, Robinson and Thompson proved the k(GV) problem for large p. This ultimately led to a complete proof of the k(GV)-problem. In this paper, we present a new proof of the k(G V)-problem for large p.

For a permutation group H on an infinite set X and a transformation f of X, let 〈f: H〉 = 〈{hfh-1:h є; H}〉 be a group closure of f. We find necessary and sufficient conditions for distinct normal subgroups of the symmetric group on X and a one-to-one transformation f of X to generate distinct group closures of f. Amongst these group closures we characterize those that are left simple, left cancellative, idempotent-free semigroups, whose congruence lattice forms a chain and whose congruences are preserved under automorphisms.

A generalization of the Pastijn product is introduced so that, on the level of e-varieties and pseudoe-varieties, this product and the regular semidirect product by completely simple semigroups ‘almost always’ coincide. This is applied to give a model of the bifree objects in every e-variety formed as a regular semidirect product of a variety of inverse semigroups by a variety of completely simple semigroups that is not a group variety.

Finite innately transitive permutation groups include all finite quasiprimitive and primitive permutation groups. In this paper, results in the theory of quasiprimitive and primitive groups are generalised to innately transitive groups, and in particular, we extend results of Praeger and Shalev. Thus we show that innately transitive groups possess parameter bounds which are similar to those for primitive groups. We also classify the innately transitive types of quotient actions of innately transitive groups.

The conjugacy classes of so-called special involutions parameterize the constituents of the action of a finite Coxeter group on the cohomology of the complement of its complexified hyperplane arrangement. In this note we give a short intrinsic characterisation of special involutions in terms of so-called bulky parabolic subgroups.

A subgroup H of a finite group G is said to be c–supplemented in G if there exists a subgroup K of G such that G = HK and H∩K is contained in coreG (H). In this paper some results for finite p–nilpotent groups are given based on some subgroups of Pc–supplemented in G, where p is a prime factor of the order of G and P is a Sylow p–subgroup of G. We also give some applications of these results.

A subgroup H of a finite G is said to be c-normal in G if there exists a normal subgroup N of G such that G = HN with H ∩ N ≤ HG = CoreG(H). We are interested in studying the influence of the c–normality of certain subgroups of prime power order on the structure of finite groups.

This paper gives a characterisation of finite supersoluble groups of Wielandt length two of order coprime to six.

The Archimedean components of triangular norms (which turn the closed unit interval into anabelian, totally ordered semigroup with neutral element 1) are studied, in particular their extension to triangular norms, and some construction methods for Archimedean components are given. The triangular norms which are uniquely determined by their Archimedean components are characterized. Using ordinal sums and additive generators, new types of left-continuous triangular norms are constructed.

In this note we characterize the abelian groups G which have two different proper subgroups N and M such that the subgroup lattice L(G)=[0, M]∪ [N, G] is the union of these intervals.

In this paper we describe the groups admitting a covering with Hall subgroups. We also determine the groups with a π1-Hall subgroup, where π1 is the connected component of the prime graph, containing the prime 2.

We give algebraic proofs of some results of Wang on homomorphisms of nonzero degree between aspherical closed orientable 3-manifolds. Our arguments apply to PDn-groups which are virtually poly-Z or have a Kropholler decomposition into parts of generalized Seifert type, for all n.

Let K be a field of prime characteristic p and let G be a finite group with a Sylow p-subgroup of order p. For any finite-dimensional K G-module V and any positive integer n, let Ln (V) denote the nth homogeneous component of the free Lie K-algebra generated by (a basis of) V. Then Ln(V) can be considered as a K G-module, called the nth Lie power of V. The main result of the paper is a formula which describes the module structure of Ln(V) up to isomorphism.

In this paper we prove that if V is a vector space over a field of positive characteristric p ≠ 5 then any regular subgroup A of exponent 5 of GL(V) is cyclic. As a consequence a conjecture of Gupta and Mazurov is proved to be true.

Let G be a finite p-solvable group for a fixed prime p. We study how certain arithmetical conditions on the set of p-regular conjugacy class sizes of G influence the p-structure of G. In particular, the structure of the p-complements of G is described when this set is {1, m, n} for arbitrary coprime integers m, n > 1. The structure of G is determined when the noncentral p-regular class lengths are consecutive numbers and when all of them are prime powers.