In the present paper we consider Fitting classes of finite soluble groups which locally satisfy additional conditions related to the behaviour of their injectors. More precisely, we study Fitting classes 1 ≠⊆such that an-injector of G is, respectively, a normal, (sub)modular, normally embedded, system permutable subgroup of G for all G ∈.

Locally normal Fitting classes were studied before by various authors. Here we prove that some important results—already known for normality—are valid for all of the above mentioned embedding properties. For instance, all these embedding properties behave nicely with respect to the Lockett section. Further, for all of these properties the class of all finite soluble groups G such that an x-injector of G has the corresponding embedding property is not closed under forming normal products, and thus can fail to be a Fitting class.

A lattice formation is a class of groups whose elements are the direct product of Hall subgroups corresponding to pairwise disjoint sets of primes. In this paper Fitting classes with stronger closure properties involving F-subnormal subgroups, for a lattice formation F of full characteristic, are studied. For a subgroup-closed saturated formation G, a characterisation of the G-projectors of finite soluble groups is also obtained. It is inspired by the characterisation of the Carter subgroups as the N-projectors, N being the class of nilpotent groups.

Characterisations of finite groups in which normality is a transitive relation are presented in the paper. We also characterise the finite groups in which every subgroup is either permutable or coincides with its permutiser as the groups in which every subgroup is permutable.

This paper is concerned with the well-known and long-standing k(G V)-problem: If the finite group G acts faithfully and irreducibly on the finite GF(p)-module V and p does not divide the order of G, is the number k(GV) of conjugacy classes of the semidirect product GV bounded above by the order of V? Over the past two decades, through the work of numerous people, by using deep character theoretic arguments this question has been answered in the affirmative except for ρ = 5 for which it is still open. In this paper we suggest a new approach to the k(G V)-problem which is independent of most of the previous work on the problem and which is mainly group theoretical. To demonstrate the potential of the new line of attack we use it to solve the k(G V)-problem for solvable G and large ρ.

Generalizing and strengthening some well-known results of Higman, B. Neumann, Hanna Neumann and Dark on embeddings into two-generator groups, we introduce a construction of subnormal verbal embedding of an arbitrary (soluble, fully ordered or torsion free) ordered countable group into a twogenerator ordered group with these properties. Further, we establish subnormal verbal embedding of defect two of an arbitrary (soluble, fully ordered or torsion free) ordered group G into a group with these properties and of the same cardinality as G, and show in connection with a problem of Heineken that the defect of such an embedding cannot be made smaller, that is, such verbal embeddings of ordered groups cannot in general be normal.

Let M be a commutative cancellative atomic monoid. We consider the behaviour of the asymptotic length functions and on M. If M is finitely generated and reduced, then we present an algorithm for the computation of both and where x is a nonidentity element of M. We also explore the values that the functions and can attain when M is a Krull monoid with torsion divisor class group, and extend a well-known result of Zaks and Skula by showing how these values can be used to characterize when M is half-factorial.

If ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

Let G be a finite group of even order, k be a field of characteristic 2, and M be a finitely generated kG-module. If M is realized by a compact G-Moore space X, then the Betti numbers of the fixed point set XCn and the multiplicities of indecomposable summands of M considered as a kCn-module are related via a localization theorem in equivariant cohomology, where Cn is a cyclic subgroup of G of order n. Explicit formulas are given for n = 2 and n = 4.

Let Γ be a free noncommutative group with free generating set A+. Let μ ∈ ℓ1(Γ) be real, symmetric, nonnegative and suppose that supp. Let λ be an endpoint of the spectrum of μ considered as a convolver on ℓ2(Γ). Then λ − μ is in the left kernel of exactly one pure state of the reduced in particular, Paschke's conjecture holds for λ − μ.

Certain permutation representations of free groups are constructed by finite approximation. The first is a construction of a cofinitary group with special properties, answering a question of Tim Wall published by Cameron. The second yields, via a method of Kepert and Willis, a totally disconnected locally compact group which is compactly generated and uniscalar but has no compact open normal subgroup. Finally, an oligomorphic group of automorphisms of the random graph is built, all of whose non-trivial subgroups have just finitely many orbits.

In 1986, Higgins proved that T(X), the semigroup (under composition) of all total transformations of a set X, has a proper dense subsemigroup if and only if X is infinite, and he obtained similar results for partial and partial one-to-one transformations. We consider the generalised transformation semigroup T(X, Y) consisting of all total transformations from X into Y under the operation α * β = αθβ, where θ is any fixed element of T(Y, X). We show that this semigroup has a proper dense subsemigroup if and only if X and Y are infinite and | Yθ| = min{|X|,|Y|}, and we obtain similar results for partial and partial one-to-one transformations. The results of Higgins then become special cases.

Davey and Quackenbush proved a strong duality for each dihedral group Dm with m odd. In this paper we extend this to a strong duality for each finite group with cyclic Sylow subgroups (such groups are known to be metacyclic).

The spherical functions of triangle buildings can be described in terms of certain two-dimensional orthogonal polynomials on Steiner's hypocycloid which are closely related to Hall-Littlewood polynomials. They lead to a one-parameter family of two-dimensional polynimial hypergroups. In this paper we investigate isotropic random walks on the vertex sets of triangle buildings in terms of their projections to these hypergroups. We present strong laws of large numbers, a central limit theorem, and a local limit theorem; all these results are well-known for homogeneous trees. Proofs are based on moment functions on hypergroups and on explicit expansions of the hypergroup characters in terms of certain two-dimensional Tchebychev polynimials.

Let and denote respectively the variety of groups of exponent dividing e, the variety of nilpotent groups of class at most c, the class of nilpotent groups and the class of finite groups. It follows from a result due to Kargapolov and Čurkin and independently to Groves that in a variety not containing all metabelian groups, each polycyclic group G belongs to . We show that G is in fact in , where c is an integer depending only on the variety. On the other hand, it is not always possible to find an integer e (depending only on the variety) such that G belongs to but we characterize the varieties in which that is possible. In this case, there exists a function f such that, if G is d-generated, then G ∈ So, when e = 1, we obtain an extension of Zel'manov's result about the restricted Burnside problem (as one might expect, this result is used in our proof). Finally, we show that the class of locally nilpotent groups of a variety forms a variety if and only if for some integers c′, e′.

Finite graphs of valency 4 and girth 4 admitting ½-transitive group actions, that is, vertex- and edge- but not arc-transitive group actions, are investigated. A graph is said to be ½-transitive if its automorphism group acts ½-transitively. There is a natural orientation of the edge set of a ½-transitive graph induced and preserved by its automorphism group. It is proved that in a finite ½-transitive graph of valency 4 and girth 4 the set of 4-cycles decomposes the edge set in such a way that either every 4-cycle is alternating or every 4-cycle is directed relative to this orientation. In the latter case vertex stabilizers are isomorphic to Z2.

For each positive integer n let N2, n denote the variety of all groups which are nilpotent of class at most 2 and which have exponent dividing n. For positive integers m and n, let N2, mN2, n denote the variety of all groups which have a normal subgroup in N2, m with factor group in N2, n. It is shown that if G ∈N2, mN2, n, where m and n are coprime, then G has a finite basis for its identities.

A locally compact semilattice with open principal filters is a zero-dimensional scattered space. Cardinal invariants of locally compact and compact semilattices with open principal filters are investigated. Structure of topological semilattices on the one-point Alexandroff compactification of an uncountable discrete space and linearly ordered compact semilattices with open principal filters are researched.

Let Λ be an ordered abelian group. It is shown that groups in a certain class can have no non-trivial action of end type on a Λ-tree. A similar result is obtained for irreducible actions.

A close connection is uncovered between the lower central series of the free associative algebra of countable rank and the descending Loewy series of the direct sum of all Solomon descent algebras Δn, n ∈ ℕ0. Each irreducible Δn-module is shown to occur in at most one Loewy section of any principal indecomposable Δn-module.A precise condition for his occurence and formulae for the Cartan numbers are obtained.

It is shown that no finite group containing a non-abelian nilpotent subgroup is dualizable. This is in contrast to the known result that every finite abelian group is dualizable (as part of the Pontryagin duality for all abelian groups) and to the result of the authors in a companion article that every finite group with cyclic Sylow subgroups is dualizable.