Throughout this paper we will assume that all groups are contained in some fixed but arbitrary universe. Thus the class of all groups becomes a set. If is a class of groups then we assume that 1 ∈ and if H ≅ G ∈ then H ∈ . German capitals will be used to denote classes of groups.

In [1] Neumann and Dey prove that the free product of two finitely generated Hopf groups is Hopf and ask whether a similar result holds for direct products. It is the purpose of this paper to show that this is not the case. We prove

THEOREM A. There exists a finitely generated group G satisfying the following conditions:

(i) G is isomorphic to a proper direct factor of itself;

(ii) G is the direct product of two Hopf groups.

In this paper we consider spaces Ap, p ∈ [1, 2], and multipliers (Ap, Aq), p ∈ [1, 2], q ∈ [1, 2]. In 4.4 and 6.1 we identify (Ap, Aq) for p ∈ [1, 2], q ∈ [p, 2], and in 7.3 we identify (A2, A1). In 7.1 we give a sufficient condition, and in 7.5 a necessary condition, for membership of (Ap, Aq), p ∈ [1, 2], q ∈ [1, 2]. We give, in 7.2, a necessary condition for membership of (A2, Aq), q ∈[l,2). We include constructive proofs of some strict inclusion results for Ap, p ∈ [1,2], (3.1 and 3.2), and also, in 5.3, for (AP,AP), p∈[l,2].

The first and still the best known computer application to groups exploits coset enumeration and this has been very thoroughly studied; see for instance [2] and [8]. No doubt this is because the algorithm is simple in the sense of programming. The Underlying mathematics is far from simple, touching as it does no logical difficulties akin to the word problem for groups, and this is reflected in the facts that random access to large tables is required and that there is no indication at any stage (for example when storage space is exhausted) whether the algorithm would be completed at any later stage. Efficient computation depends on choosing a subgroup of small index m in the group under examination, for group elements will be represented as permutations of degree m, and the larger m is the more tedious it will be to check properties like orders of group elements. Yet in many cases m may have to be fairly large so that the subgroup is “corefree” i.e. the representation is faithful.

A subgroup H of a group G is said to be subnormal in G if there is a finite chain of subgroups, each normal in its successor, connecting H to G. If such chains exist there is one of minimal length; the number of strict inclusions in this chain is called the subnormal index, or defect, of H in G. The rather large class of groups which have an upper bound for the subnormal indices of their subnormal subgroups has been inverstigated to same extent, mainly with a restriction to solublegroups — for instance, in [10] McDougall considered soluble p-groups in this class. Robinson, in [14], restricted his attention to wreath products of nilpotent groups but extended his investigations to the strictly larger class of groups in which the intersection of any family of subnormal subgroups is a subnormal subgroup. These groups are said to have the subnormal intersection property.

LetG be a finite group, and k a field of finite characteristic p, such that the polynomial x¦G¦ –1 splits completely in k[x]. Let Β be a kG-block which has defect group D which is cylclic of order pd (d ≧ 1). Brauer showed in a famous paper [2] that, in case d = 1, the decomposition matrix of Β is determined by a certain positive integer e which divides p − 1, and a tree Г, a connected acyclic linear graph of e + 1 vertices and e edges. Twenty-five years later Dade ([3]) extended Brauer's theorem to the general case.

Almost periodic solutions of a first order almost periodic differential equation in Rp are shown to have less than p basic frequencies additional to the basic frequencies of the almost periodic right hand of the equation.

1. In their fundamental paper of 1949, Higman, Neumann and Neumann proved for the first time that a countable group can always be embedded in some 2-generator group: [1], Theorem IV. Two kinds of improvement of this result have recently appeared. In [4], Theorem 2, Dark has shown that the embedding can always be made subnormally. On the other hand, in [2], Theorem 2.1, Levin has shown that the two generators can be given preassigned orders m > 1 and n > 2; and in [3], Miller and Schupp prove that the 2-generator group can also be made to satisfy several additional requirements, such as being complete and Hopfian.

For sums of independent and identically distributed random variables xn, the HartmanWintner law of the iterated logarithm is equivalent to xn ∈ L2. We show that this is also true when the xn, form a stationary, ergodic martingale difference sequence. This is accomplished by extending a theorem of Volker Strassen to the present context.

In 1904 Poincaré [11] conjectured that every compact, simply-connected closed 3-dimensional manifold is homeomorphic to a 3-sphere. The corresponding result for dimension 2 is classical; for dimension ≧ 5 it was proved by Smale [12] and Stallings [13], but for dimensions 3 and 4 the question remains open. It has been discovered in recent years that the 3-dimensional Poincaré conjecture could be reformulated in purely algebraic terms [6, 10, 14, 15] however the algebraic problems which are posed in the references cited above have not, to date, proved tractable.

In this note we apply a particular technique to obtain information on the homology homomorphism ε*: H*(G; A) → H* (Q; A) associated with a group extension and a Q-module A. The technique consists of using ε itself to pull-back (0.1); that is, we construct the pull-back extension induced from (0.1) by ε. This, however, is nothing but the semidirect product, NG, of N and G, with G operating on the left on N by conjugation. Thus we obtain from (0.1) the commutative diagram where ε1 is the projection and et is the multiplication ε1(n, x) = nx,n∈N,x∈G. We now apply the Lyndon-Hochschild-Serre spectral sequence functor to (0.2) and carry out computations in dimensions 2 and 3.

This paper is a study of nonsingular rings with essential socles. These rings were first investigated by Goldie [5] who studied the Artinian case and showed that an indecomposable nonsingular generalized uniserial ring is isomorphic to a full blocked triangular matrix ring over a sfield. The structure of nonsingular rings in which every ideal generated by a primitive idempotent is uniform was determined for the Artinian case by Gordon [6] and Colby and Rutter [2], and for the semiprimary case by Zaks [12]. Nonsingular rings with essential socles and finite identities were characterized by Gordon [7] and the author [10]. All these results were obtained by representing the rings in question as matrix rings. In this paper a matrix representation of arbitrary nonsingular rings with essential socles is found (section 2). The above results are special cases of this representation. A general method for representing rings as matrices is developed in section 1.

Any variety of groups is generated by its free group of countably infinite rank. A problem that appears in various forms in Hanna Neumann's book [7] (see, for intance, sections 2.4, 2.5, 3.5, 3.6) is that of determining if a given variety B can be generated by Fk(B), one of its free groups of finite rank; and if so, if Fn(B) is residually a k-generator group for all n ≧ k. (Here, as in the sequel, all unexplained notation follows [7].)

This paper presents a basis for the laws which hold in each of the finite simple groups, PSL(2, 2n), n> ≥ 2, thus partially solving a problem raised by Cossey, Macdonald and Street [3]. They considered the more general problem of finding bases for the laws which hold in PSL(2, pn), and succeded in finding a number of general laws, and in completing bases for pn ≤ 11. The solution of the general problem appears to be very difficult.

This note gives a way of looking at Malcev conditions for varieties as ideals in a certain lattice. Though this viewpoint (so far) yields no new results, we feel it puts Walter Taylor's results [3] characterizing “Malcev definable classes of varieties” into a clearer perspective and is therefore worth mentioning. The author is grateful to R. Wille for pointing out W. Taylor's paper to him.

A weak but still unsolved version of the first half of Problem 7 in Hanna Neumann's book [5] asks whether the product of a nontrivial join-irreducible variety and an arbitrary variety is necessarily also join-irreducible. One of the results of this paper is a positive answer provided either is abelian or the infiniterank free groups of have no nontrivial abelian verbal subgroups. For a general discussion, see [4]; all unexplained notation is as in [5].

In this paper we are going to prove the following results:

Theorem 1. There exist two bounded hopfian lattices such that their {0,1}- free product is not hopfian.

Theorem 2. There exist two hopfian lattices such that their free product is not hopfian.

In Theorem 2 free product (coproduct, sum) has its usual meaning (see, for instance, [4]); in Theorem 1 we use the usual definition but all lattices are assumed to be bounded (that is, having a least element 0 and largest element 1) and all homomorphisms are assumed to be {0, l}-homomorphisms (that is, homomorphisms preserving 0 and 1).

The results in this paper are consequences of an attempt many years ago to extend to loops some form of the theorem of Lyndon [12] that any nilpotent group has finitely based identities. Having failed in this, we looked for other algebras for which a similar approach might work. The algebra has to belong to a variety in which finitely generated algebras are finitely related and we must be able to bound the number of variables needed in a basis. Commutative Moufang loops, because of the extensive commutator calculus available (Bruck, [4]), provide one example (Evans, [6]). Here we give two examples from rings, namely associative rings satisfying xn = x (more generally, satisfying an identity x2 · p(x) = x) and nilpotent (non-associative) rings. We are also able to extend some results of Higman [9] on product varieties and we show that for associative rings the product of a nilpotent variety and a finitely based bariety is finitely based.