A Neumann subgroup of the classical modular group is by definition a complement of a maximal parabolic subgroup. Recently Neumann subgroups have been studied in a series of papers by Brenner and Lyndon. There is a natural extension of the notion of a Neumann subgroup in the context of any finitely generated Fuchsian group Γ acting on the hyperbolic plane H such that Γ/H is homeomorphic to an open disk. Using a new geometric method we extend the work of Brenner and Lyndon in this more general context.

A group G is said to be conjugacy p-separable if two non-conjugate elements of G remain non-conjugate in some finite p-group endomorphic image of G. We show that the non-cyclic free centre-by-metabelian groups are not conjugacy p-separable for any prime p. On the other hand, we show that every free centre-by-metabelian group has the solvable conjugacy problem

It has been shown by one of the authors that the system of idempotents of monoids on a group G of Lie type with Dynkin diagram Γ can be classified by the following data: a partially ordered set U with maximum element 1 and a map λ: U → 2Γ with λ(1) = Γ and with the property that for all J1, J2, J3 ∈ U with J1 > J2 > J3, any connected component of λ(J2) is contained in either λ(J1) or λ(J3). In this paper we show that λ comes from a regular monoid if and only if the following conditions are satisfied: (1) U is a ∧-semilattice; (2) If J1, J2 ∈ U, then λ(J1)∧ λ(J2) λ(J1 ∧ J2); (3) If θ ∈ Γ, J ∈ U, then max{J1 ∈ U|J1 > J, θ ∈ λ (J1)} exists; (4) If J1, J2 ∈ U with J1 > J2 and if X is a two element discrete subset of λ(J1) ∪ λ(J2), then X λ(J) for some J ∈ UJ with J1 > J > J2.

In this paper, it is shown that any connected, small category can be embedded in a semi-groupoid (a category in which there is at least one isomorphism between any two elements) in such a way that the embedding includes a homotopy equivalence of classifying spaces. This immediately gives a monoid whose classifying space is of the same homotopy type as that of the small category. This construction is essentially algorithmic, and furthermore, yields a finitely presented monoid whenever the small category is finitely presented. Some of these results are generalizations of ideas of McDuff.

All subnormal subgroups of hypernormalizing groups have by definition subnormal normalizers. It is shown that finite soluble HN-groups belong to the class of groups of Fitting length three. Finite HN-groups are considered including those with subnormal quotient isomorphic to SL(2,5).

In this note, for any given simple group obtained from an orthogonal or unitary group of non-zero index, by a procedure similar to the construction of Chevalley groups and twisted groups, we construct a simple group which is identified with the given simple classical group. The simple groups constructed in this note can be interpreted as generalized simple groups of Lie type. Thus all simple groups of Lie type of types An, Bn, Cn and Dn and all generalized simple groups of Lie type constructed in this note exhaust all simple classical groups with non-zero indices.

A construction for Fitting formations given by the author and C. L. Kanes is generalised. The original examples were based on the use of Fitting families of modules over algebraically closed fields. An example of Haberl and Heineken in 1984 suggested that the methods should work with modules over arbitrary fields. We show that this is indeed the case, provided we restrictthe class of groups considered.

In this paper we continue our investigations of a construction method for subnear-rings of M(G) proposed by H. Wielandt. For a meromorphic product H, H ⊂ Gk, G finite, we obtain necessary and sufficient conditions for M(G, k, H) to be a near-field.

In this note we present a general Jordan-Hölder type theorem for modular lattices and apply it to obtain various (old and new) versions of the Jordan-Hölder Theorem for finite groups.

We determine the structure of a nonabelian group G of odd order such that some automorphism of G sends exactly (1/p)|G| elements to their cubes, where p is the smallest prime dividing |G|. These groups are close to being abelian in the sense that they either have nilpotency class 2 or have an abelian subgroup of index p.

In this note we introduce a self-centralizing characteristic subgroup, associated with quasinilpotent injectors, of a finite group.

Let G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.

Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

In order to classify solvable groups Philip Hall introduced in 1939 the concept of isoclinism. Subsequently he defined a more general notion called isologism. This is so to speak isoclinism with respect to a certain variety of groups. The equivalence relation isologism partitions the class of all groups into families. The present paper is concerned with the internal structure of these families.

Let R be a not necessarily commutative local ring, M a free R-module, and π ∈ GL(M) such that B(π) = im(π –1)is a subspace of M. Then π = σ1…σtρ, where σi are simple mappings of given types, ρ is a simple mapping, B(sgr;i) and B(ρ) are subspaces and t ≤ dim B(π).

Certain central products of the binary polyhedral groups with finite cyclic groups are here shown to have presentations with two generators and two defining relations; this disproves a conjecture of the second author, stated in J. Austral. Math. Soc. Ser. A 38 (1985), 230–240.

A Σ-group is an abelian group on which is given a family of infinite sums having properties suggested by, but weaker than, those which hold for absolutely convergent series of real or complex numbers. Two closely related questions are considered. The first concerns the construction of a Σ-group from an arbitrary abelian group on which certain series are given to be summable, certain of these series being required to sum to zero. This leads to a Σ-theoretic construction of R from Q and in general of the completion of an arbitrary metrizable abelian group (with the associated unconditional sums) from that group. The second question is whether, in a given Σ-group, the values of the infinite sums may be determined solely from a knowledge of which series are summable. Such a Σ-group is said to be relatively free and it is shown that all metrizable abelian groups are relatively free.

This paper constructs a minimal faithful representation of a semilattice of groups by partial transformations. The solution is expressed in terms of join irreducible elements of the semilattice and minimal faithful representations of groups with respect to certain normal subgroups.

This is a study of formal power series under the binary operation of formal composition from a group-theoretical point of view. Various “large” properties are derived.

The following questions are studied: When is a semigroup graded ring left Noetherian, respectively semiprime left Goldie? Necessary sufficient conditions are proved for cancellative semigroup-graded subrings of rings weakly or strongly graded by a polycyclic-by-finite (unique product) group. For semigroup rings R[S] we also give a solution to the problem in case S is an inverse semigroup.

In this paper we show that a group A is embedded in any finite group G as a subnormal subgroup with low degree of complication, provided that the automorphism group of A satisfies a condition depending on some Fitting class (which coincides with completeness for the Fitting class of all groups). A criterion is given for these groups as to whether they can be embedded subnormally in the commutator subgroup of some finite group or not.