Following a suggestion of G. Higman we say that the group G is SQ-universal if every countable group is embeddable in some factor group of G. It is a well-known theorem of G. Higman, B. H. Neumann and Hanna Neumann that the free group of rank 2 is sq-universal in this sense. Several different proofs are now available (see, for example, [1] or [9]). It is my intention to prove the LEmma. If H is a subgroup of finite index in a group G, then G is SQ-universal if and only if H is SQ-universal.

In her paper [3], Osofsky exhibited an example of a ring R containing 16 elements which (i) is equal to its left complete ring of quotients, (ii) is not self-injective and (iii) whose injective hull HR = H(RR) allows a ring structure extending the R-module structure of HR. In the present note, we offer a general method of constructing such rings; in particular, given a non-trivial split Frobenius algebra A and a natural n ≧ 2, a certain ring of n x n matrices over A provides such an example. Here, taking for A the semi-direct extension of Z/2Z by itself and n = 2, one gets the example of Osofsky. Thus, our approach answers her question on finding a non-computational method for proving the existence of such rings.

Combinatorial group theory has roots in Poincaré's work on the topology of manifolds, which in turn was based on problems in differential equations and analytic number theory. Thus the Fuchsian groups, which are the fundamental (first homotopy) groups of oriented negatively curved compact surfaces, served as important models in their day. In the last few years there have been advances in the understanding of the structure of fundamental groups of negatively curved manifolds, some of them based on examples from analytic number theory. Here I describe one of these developments and pose a few difficult combinatorial questions.

The object of this note is to record a property of finite, perfect, centrally closed groups, where, by definition, G si centrally closed if and only if whenever E/Z ≅ G and Z ⊆ E' ∩ Z(E), then Z = 1.

In this paper the following notation will be used: for any group G, positive integer c and non-negative integer n, Gc is the cth term of the lower central series of G and δ nGc is the nth term of the derived series of Gc.

The objective of this paper is to prove the following generalization of the main result in [2]: THeorem. Let G be a finite simple group which possesses an involution t such that the centralizer of t in G is isomorphic to the centralizer of an involution in H. Then G is isomorphic to L5(2), M24, or H.

Problem 11 of Hanna Neumann's book [3] asks whether the product variety Β4Β2 has a finite basis for its laws. (For any positive integer k, Βk denotes the variety of all groups of exponent diving k.) I think that Β4Β2 was being suggested as a plausible canditate for a variety without the finite basis property; of course, at a time when no such example was known. It is the primary object of this note to verify the fact that Β4Β2 is not finitely based. Β4Β2 provides, therefore, probably the simplest example known at present of a variety which is not finitely based.

A universal power automorphism (Cooper [1]) of a group is an automorphism mapping every element x to a power xn for some fixed integer n. It is long known that a group admitting such an automorphism with n= −1, 2 or 3 must be Abelian. Miller [5] showed that for every other non-zero integral value of n there exist non-Abelian groups admitting a non-trivial universal power automorphism x→xn.

Let L be a Lie algebra over the field F. A lattice automorphism of ℒ is an automorphism φ: ℒ(L) → ℒ(L) of the lattice ℒ(L) of all subalgebras of L. We seek to describe the lattice automorphisms in terms of maps σ: L→L of the underlying algebra. A semi-automorphism σ of L is an automorphism of the algebraic system consisting of the pair (F, L), that is, a pair of maps σ: F→ F, σ: L→ L preserving the operations. Thus σ: F → F is an automorphism of F and (x + y)σ = xσ + yσ, (xy)σ = xσyσ (λx)σ = λσxσ for all x, y ε L and λ ε F. Clearly, any semi-automorphism of L induces a lattice automorphism. To study a given lattice automorphism φ, we select a semi-automorphism a such that φσ-1 fixes certain subalgebras, and so we reduce the problem to the investigation of lattice automorphisms leaving these subalgebras fixed.

In this note “vector space” will mean “Banach space” unless otherwise specified. Accordingly “Lie algebra” will stand for “Banach Lie algebra”. Morphisms between Lie algebras will be assumed continuous. A Banach algebra B will be always assumed associative, and it will be also viewed as a Lie algebra with product [X, YXY− YX. In particular, the Lie algebra gl(V) of endomorphisms of a vector space V will be equipped with the uniform norm. A morphism of Lie algebras L → gl(V) will b called a representation of L in gl(V). Also, if B is a Banach algebra, a morphism of Lie algebras L → B will be called a representation of L in B. From such one evidently obtains a representation of L in gl(B). A representation will be called faithful if it is injective.

Let be the given trigonometric series, then the formal product of them is defined by where the last series is supposed to be convergent for every n.

The study of minimal irregular p-groups was initiated by Mann [9]. He defines such a group to be a finite irregular p-group in which every proper section is regular. He then deduces a large number of properties of such groups and shows, by construction, that there is no bound on their exponent. In this note methods involving varieties of groups are used to show that all minimal irregular p-groups can be ‘built up’ from cyclic groups, groups of exponent p and a relatively small class of metabelian minimal irregular p-groups. To phrase this more precisely, say that a group P is derivable from a family of groups {Qi} if P is a homomorphic image of a subdirect product of the Qi. Then we will prove as Theorem 2.2: a minimal irregular p-group of exponent pn and nilpotency class c is derivable from a family of three groups: i) a cyclic group of order pn, ii) a 2-generator monolithic group of exponent p and class c, iii) a minimal irregular p-group whi h is metabelian, has class p and has exponent p2.

A subgroup H of a group G is said to be a permutablesubgroup of G if HK = KH 〈H, K〉 for all subgroups K of G. It is known that a core-free permutable subgroup H of a finite group G is always nilpotent [5]; and even when G is not finite, H is always a subdirect product of finite nilpotent groups [11]. Thus nilpotency is a measure of the extent to which a permutable subgroup differs from being normal. Examples of non-abelian, core-free, permutable subgroups are rare and difficult to construct. The first, due to Thompson [12], had class 2. Further examples of class 2 appeared in [8]. More recently Bradway, Gross and Scott [1] have constructed corresponding to each positive integer c and each prime p ≥ c, a finite p-group possessing a core-free permutable subgroup of class c. In [3] Gross succeeded in dispensing with the requirement p ≥ c.

In 1961 Graham Higman [1] proved that a finitely generated group is a subgroup of a finitely presented group if, and only if, it is recursively presented. Therefore a finitely generated metabelian group can be embedded in a finitely presented group.

The theory of alternating bilinear forms on finite dimensional vector spaces V is well understood; two forms on V are equivalent if and only if they have equal ranks. The situation for alternating trilinear forms is much harder. This is partly because the number of forms of a given dimension is not independent of the underlying field and so there is no useful canonical description of an alternating trilinear form.