In a paper published in this journal [1], J. T. Buckely, J. C. Lennox, B. H. Neumann and the authors considered the class of CF-groups, that G such that |H: CoreG (H)| is finite for all subgroups H. It is shown that locally finite CF-groups are abelian-by-finite and BCF, that is, there is an integer n such that |H: CoreG(H)| ≤ n for all subgroups H. The present paper studies these properties in the class of locally graded groups, the main result being that locally graded BCF-groups are abelian-by-finite. Whether locally graded CF-groups are BFC remains an open question. In this direction, the following problems is posed. Does there exist a finitely generated infinite periodic residually finite group in which all subgroups are finite or of finite index? Such groups are locally graded and CF but not BCF.

A group variety defined by one semigroup law in two variables is constructed and it is proved that its free group is not a periodic extension of a locally soluble group.

We show that the growth function of a finitely generated linear semigroup S ⊆ Mn(K) is controlled by its behaviour on finitely many cancellative subsemigroups of S. If the growth of S is polynomially bounded, then every cancellative subsemigroup T of S has a group of fractions G ⊆ Mn (K) which is nilpotent-by-finite and of finite rank. We prove that the latter condition, strengthened by the hypothesis that every such G has a finite unipotent radical, is sufficient for S to have a polynomial growth. Moreover, the degree of growth of S is then bounded by a polynomial f(n, r) in n and the maximal degree r of growth of finitely generated cancellative T ⊆ S.

We study commutators in pseudo-orthogonal groups O2nR (including unitary, symplectic, and ordinary orthogonal groups) and in the conformal pseudo-orthogonal groups GO2nR. We estimate the number of commutators, c(O2nR) and c(GO2nR), needed to represent every element in the commutator subgroup. We show that c(O2nR) ≤ 4 if R satisfies the ∧-stable condition and either n ≥ 3 or n = 2 and 1 is the sum of two units in R, and that c(GO2nR) ≤ 3 when the involution is trivial and ∧ = R∈. We also show that c(O2nR) ≤ 3 and c(GO2nR) ≤ 2 for the ordinary orthogonal group O2nR over a commutative ring R of absolute stable rank 1 where either n ≥ 3 or n = 2 and 1 is the sum of two units in R.

A group G has all of its subgroups normal-by-finite if H / coreG(H) is finite for all subgroups H of G. These groups can be quite complicated in general, as is seen from the so-called Tarski groups. However, the locally finite groups of this type are shown to be abelian-by-finite; and they are then boundedly core-finite, that is to say, there is a bound depending on G only for the indices | H: coreG(H)|.

That the monoid of all transformations of any set and the monoid of all endomorphisms of any vector space over a division ring are regular (in the sense of von Neumann) has been known for many years (see [6] and [16], respectively). A common generalization of these results to the endomorphism monoid of an independence algebra can be found in [13]. It also follows from [13] that the endomorphism monoid of a free G-act is regular, where G is any group. In the present paper we use a version of the wreath product construction of [8], [9] to determine the projective right S-acts (S any monoid) whose endomorphism monoid is regular.

The aim of this paper is to consider Problem 1 posed by Stewart and Wiegold in [6]. The main result is that if G is a finitely generated perfect group having non-trivial finite images, then there exists a finite image B of G such that the growth sequence of B is eventuallly faster than that of every finite image of G. Moreover we investigate the growth sequences of simple groups of the same order.

A closure operation connected with Hall subgroups is introduced for classes of finite soluble groups, and it is shown that this operation can be used to give a criterion for membership of certain special Fitting classes, including the so-called ‘central-socle’ classes.

We give an improved bound for the order of a p-group enjoying the title property. We also point out relations between the upper central series of a p-group and of its maximal subgroups.

A common generalization of the author's embedding theorem concerning the E-unitary regular semigroups with regular band of idempotents, and Billhardt's and Ismaeel's embedding theorem on the inverse semigroups, the closure of whose set of idempotents is a Clifford semigroup, is presented. We prove that each orthodox semigroup with a regular band of idempotents, which is an extension of an orthogroup K by a group, can be embedded into a semidirect product of an orthogroup K′ by a group, where K′ belongs to the variety of orthogroups generated by K. The proof is based on a criterion of embeddability into a semidirect product of an orthodox semigroup by a group and uses bilocality of orthogroup bivarieties.

In this paper we shall extend the classical theory of Morita equivalence to semigroups with local units. We shall use the concept of a Morita context to rediscover the Rees theorem and to characterise completely 0-simple and regular bisimple semigroups.

A number of constructions are given for arc-transitive digraphs, based on modifications of permutation representations of finite groups. In particular, it is shown that for every positive integer s and for any transitive permutation group p of degree k, there are infinitely many examples of a finite k-regular digraph with a group of automorphisms acting transitively on s-arcs (but not on (s + 1)-arcs), such that the stabilizer of a vertex induces the action of P on the out-neighbour set.

Group actions on ℝ-trees may be split into different types, and in Section 1 of this paper five distinct types are defined, with one type splitting into two sub-types. For a group G acting as a group of isometries on an ℝ-tree, conditions are considered under which a subgroup or a factor group may inherit the same type of action as G. In Section 2 subgroups of finite index are considered, and in Section 3 normal subgroups and also factor groups are considered. The results obtained here, Theorems 2.1 and 3.4, allow restrictions on possible types of actions for hypercentral, hypercyclic and hyperabelian groups to be given in Theorem 3.6. In Section 4 finitely generated subgroups are considered, and this gives rise to restrictions on possible actions for groups with certain local properties. The results throughout are stated in terms of group actions on trees. Using Chiswell's construction in [3], they could equally be stated in terms of restrictions on possible types of Lyndon length functions.

Wielandt [4] has shown that a common subnormal subgroup of two permutable subgroups of finite group is subnormal in their product. When G is infinite it seems unlikely that Wielandt's theorem will still be true, but an example illustrating this appears to be difficult, even if G is an FC-group (that is groups in which each element has only finitely many conjugates; see [2]). However, if we replace subnormality by ascendancy we have the following.

In [7] S. Pride gave a family of examples of finitely presented groups of cohomological dimension 2 having no non-trivial action on a simplicial tree. We show here that his examples have no non-trivial action on a Λ-tree, for any ordered abelian group Λ. This provides further slight evidence for an affirmative answer to Question A in §3.1 of [8]. We also give another similar family of examples.

If R is a 2-group of symplectic type with exponent 4, then R is isomorphic to the extraspecial group , or to the central product 4 o 21+2n of a cyclic group of order 4 and an extraspecial group, with central subgroups of order 2 amalgamated. This paper gives an explicit description of a projective representation of the group A of automorphisms of R centralizing Z(R), obtained from a faithful representation of R of degree 2n. The 2-cocycle associated with this projective representation takes values which are powers of −1 if R is isomorphic to and powers of otherwise. This explicit description of a projective representation is useful for computing character values or computing with central extensions of A. Such central extensions arise naturally in Aschbacher's classification of the subgroups of classical groups.

We give an alternative short proof of a recent theorem of J. A. Hillman and P.A. Linnell that an elementary amenable group with finite Hirsch number has, modulo its locally finite radical, a soluble normal subgroup with index and derived length bounded only in terms of the Hirsch number of the group.

A balanced directed cycle design with parameters (υ, k, 1), sometimes called a (υ, k, 1)-design, is a decomposition of the complete directed graph into edge disjoint directed cycles of length k. A complete classification is given of (υ, k, 1)-designs admitting the holomorph {øa, b: x ↦ ax + b∣ a, b ∈ Zυ, (a, υ1) = 1} of the cyclic group Zυ as a group of automorphisms. In particular it is shown that such a design exists if and ony if one of (a) k = 2, (b) p ≡ 1 (mod k) for each prime p dividing υ, or (c) k is the least prime dividing υ, k2 does not divide υ, and p ≡ 1 (mod k) for each prime p < k dividing υ.

The main result of this paper is an upper bound for the number of maximal subgroups in finite solvable groups. Our result improves an earlier one of Cook, Wiegold and Williamson [1]. At the end, we use our bound to deduce an estimation for the total number of subgroups in finite solvable groups.

In this paper conditions of M-symmetry, strong, semimodularity and θ-modularity for the congruence lattice L (S) of a regular ω-semigroup S are studied. They are proved to be equivalent to modularity. Moreover it is proved that the kernel relation is a congruence on L(S) if and only if L(S) is modular, generalizing an analogous result stated by Petrich for bisimple ω-semigroups.