This paper is devoted to a study of pseudocomplements in groupoids. A characterization of an intraregular groupoid is obtained in terms of prime ideals. It is proved that the set of dense elements of an intraregular groupoid S with 0 is the intersection of all the maximal filters of S and that the set of normal elements of an intraregular groupoid closed for pseudocomplements forms a Boolean algebra under natural operations. It is shown that the pseudocomplement of an ideal of an intraregular groupoid with 0 is the intersection of all the minimal prime ideas not containing it.

Ore (1942) studied the automorphisms of finite monomial groups and Holmes (1956, pp. 23–93) has given the form of the automorphisms of the restricted monomial groups in the infinite case. The automorphism group of a standard wreath product has been studied by Houghton (1962) and Segal (1973, Chapter 4). Monomial groups and standard wreath products are both special cases of permutational wreath product. Here we investigate the automorphisms of the permutational wreath product and consider to what extent the results holding in the special cases remain true for the general construction. Our results extend those of Bunt (1968).

Let G be a doubly transitive permutation group on a finite set Ω, and let Kα be a normal subgroup of the stabilizer Gα of a point α in Ω. If the action of Gα on the set of orbits of Kα in Ω − {α} is 2-primitive with kernel Kα it is shown that either G is a normal extension of PSL(3, q) or Kα ∩ Gγ is a strongly closed subgroup of Gαγ in Gα, where γ ∈ Ω − {α}. If in addition the action of Gα on the set of orbits of Kα is assumed to be 3-transitive, extra information is obtained using permutation theoretic and centralizer ring methods. In the case where Kα has three orbits in Ω − {α} strong restrictions are obtained on either the structure of G or the degrees of certain irreducible characters of G. Subject classification (Amer. Math. Soc. (MOS) 1970: 20 B 20, 20 B 25.

A semigroup over a generalized tree, denoted by the term ℳL-semigroup, is a compact semigroup S such that Green's relation H is a congruence on S and S/H is an abelian generalized tree with idempotent endpoints and E(S/H) a Lawson semilattice. Each such semigroup is characterized as being constructible from cylindrical subsemigroups of S and the generalized tree S/H in a manner similar to the construction of semigroups over trees and of the hormos. Indeed, semigroups over trees are shown to be particular examples of the construction given herein.

Let G be a finite group with d(G) = α, d(G/G′) = β≥1. If G has non-abelian simple images, let s denote the order of a smallest such image. Then d(Gn) = βn provided that βn≥α + 1 + log8n. If all simple images of G are abelian, then d(Gn) = βn provided that βn≥α. If G is non-trivial and perfect, with s again denoting the order of a smallest non-abelian simple image, then d(Gsn)≼d(G) + n for all n≥0. These results improve on results in previous papers with similar titles.

The product of two subsets C, D of a group is defined as . The power Ce is defined inductively by C0 = {1}, Ce = CCe−1 = Ce−1C. It is known that in the alternating group An, n > 4, there is a conjugacy class C such that CC covers An. On the other hand, there is a conjugacy class D such that not only DD≠An, but even De≠An for e<[n/2]. It may be conjectured that as n ← ∞, almost all classes C satisfy C3 = An. In this article, it is shown that as n ← ∞, almost all classes C satisfy C4 = An.

Given a group G, we may ask whether it is the commutator subgroup of some group G. For example, every abelian group G is the commutator subgroup of a semi-direct product of G x G by a cyclic group of order 2. On the other hand, no symmetric group Sn(n>2) is the commutator subgroup of any group G. In this paper we examine the classical linear groups over finite fields K of characteristic not equal to 2, and determine which can be commutator subgroups of other groups. In particular, we settle the question for all normal subgroups of the general linear groups GLn(K), the unitary groups Un(K) (n≠4), and the orthogonal groups On(K) (n≧7).

A sandwich semigroup of continuous functions consists of continuous functions with domains all in some space X and ranges all in some space Y with multiplication defined by fg = foαog where α is a fixed continuous function from a subspace of Y into X. These semigroups include, as special cases, a number of semigroups previously studied by various people. In this paper, we characterize the regular elements of such semigroups and we completely determine Green's relations for the regular elements. We also determine the maximal subgroups and, finally, we apply some of these results to semigroups of Boolean ring homomorphisms.

If is a saturated formation of finite soluble groups and G is a finite group whose -residual A is abelian then it is well known that G splits over A and the complements are conjugate. Hartley and Tomkinson (1975) considered the special case of this result in which is the class of nilpotent groups and obtained similar results for abelian-by-hypercentral groups with rank restrictions on the abelian normal subgroup. Here we consider the super-soluble case, obtaining corresponding results for abelian-by-hypercyclic groups.

In this paper the concept of length as defined for groups by Lausch–Nöbauer in their book Algebra of Polynomials (North Holland, Amsterdam, 1973) is generalized in several ways. It turns out that the main results of Lausch-Nöbauer concerning it remain valid for this generalization.

Let ℝ∞ be the direct limit of the Euclidean spaces ℝn. Now the orthogonal group O(∞) acts on ℝn and the direct limit O(∞) of the groups O(∞) acts on ℝ∞. The infinite pin group Pin(∞) is an extension of ℤ2 by O(∞) and admits the following presentation: the generators are the unit vectors xf in ℝ∞ and the relations are

A length function, for a group, associates to an element x a real number |x| subject to certain axioms, including a cancellation axiom which embodies certain cancellation properties for elements of a free group. Integer valued length functions were introduced by Roger Lyndon [1] where, with a more restrictive set of axioms than ours, it is shown that a length function for a group is given by a restriction of the usual length function on some free product.

In this paper we continue our investigation of the topological filtration on the complex representation ring R(G) of a finite group, see [4] and [5]. To recall the basic definitions from (1): let

map a k-dimensional representation ζ to the (flat) vector bundle over the classifying space BG associated to the universal G-bundle by the G-structure on Ck. Then, if denotes the (2k − l)-skeleton of BG,

In this note I settle a question which arose out of my first paper under the above title (cf. [1]), where I considered the classgroup C(Z(Γ)) of the integral groupring Z(Γ) of a finite Abelian group Γ. This classgroup maps onto the classgroup C() of the maximal order of the rational groupring Q(Γ), and C() is the product of the ideal classgroups of the algebraic number fields which occur as components of Q(Γ) and is thus in a sense known. One is then interested in the kernel D(Z(Γ)) of C(Z(Γ)) → C() and in its order k(Γ). In [1] I proved that, for Γ a p-group, k(Γ) is a power of p. I also computed k(Γ) for small exponents. My computation used crucially the fact that, for the groups Γ considered, the groups of units of algebraic integers which occurred were finite, i.e. that the only number fields which turned up were Q and Q(n) with n4 = 1 or n6 = 1. The numerical results obtained led me to the question whether in fact k(Γ) tends to infinity with the order of Γ.