The Fitting class of finite soluble π-groups, where π is an arbitrary set of primes, has the property that each complement of an -avoided, complemented chief factor of any finite soluble group G contains an -injector of G. In other words, each -avoided, complemented chief factor of G is -complemented in the sense of Hartley (see [2]).

We shall take for granted the basic terminology currently in use in the theory of varieties of groups. Kovács, Newman, Pentony [2] and Levin [3] prove that if m is an integer greater than 2, then the variety Νm of all nilpotent groups of class at most m is generated by its free group Fm-1(Νm) of rank m – 1 but not by its free group Fm–2(Νm) of rank m — 2. That is, the free groups Fk(Nm), 2≦k ≦ m – 2, do not generate Nm. In general little is known of the varieties generated by them. The purpose of the present paper is to record the varieties of the free groups Fk(Nm) of the nilpotent varieties Nm of all nilpotent groups of class at most m for 2 ≦ k ≦ m – 2 and 5 ≦ m ≦ 6. This is done by describing a basis for the laws in these groups, that is a set of laws the fully invariant closure of which is the set of all laws for Fk(Nm). The set of laws, which, together with the appropriate nilpotency law, form a basis for the relevant groups Fk(Nm) are listed below: .

Fellowship of the Australian Academy of Science and Fellowship of the Australian College of Education are formal recognition of Hanna Neumann's impact on a country she had first set foot in only in August 1963. But then`Hanna Neumann was a remarkable person. Throughout her life she had won the love and respect of many people. The extent of this can not really be measured, however some indications can be given. A memorial meeting held in Canberra overflowed a large lecture theatre even though it was virtually vacation time. A collection of papers dedicated to her memory and a fund to provide some form of memorial to her have both drawn quite overwhelming support from many parts of the world. One finds a tremendous list of words describing her memorable qualities: warm, enthusiastic, inspiring, energetic, firm, tactful, sympathetic, efficient, patient, shrewd, humble, peace-loving, courageous, gracious, … No words can hope to evoke more than a pale shadow of such a person; this story must be read in such a light.

An exponential polynomial is a finite linear combination of terms unea: t→tneat where n is any non-negative integer and a is any complex number. The set X of exponetial polynomials is clearly a vector space over the field of complex numbers C and this set is identical with the set of solutions to all homogeneous linear ordinary differential equations with constant coefficients.

Magill, Jr. and Yamamuro have been responsible in recent years for a number of papers showing that the property that every automorphism is inner is held by many semigroups of functions and relations on topological spaces. Following [9], we say a semigroup has the Magill property if every automorphism is inner. we say a semigroup has the Magill property if every automorphism is inner. That the semigroup of Fréchet differentiable selfmaps, D of a finite dimensional Banach space E, had the Magill property was shown in [10], while a lengthy result in [6] extended this to the semigroup of k times Fréchet differentiable selfmaps, Dk, of a Fréchet Montel space (FM-space). In the latter paper it was noted that with a little additional effort the semigroup Ck, of k times continuously Fréchet differentiable selfmaps of FM-space, could be shown to possess the Magill property. It is the purpose of this paper to present a simpler proof of this result in the case where the underlying space is finite dimensional.

Let C[[t]] denote the algebra of all formal power series over the complex field C in a commutative indeterminate t with the weak topology determined by the projections pj: Σαiti ↦αj. A subalgebra A of C[[t]] is a Banach algebra of power series if it contains the polynomials and is a Banach algebra under a norm such that the inclusion map A ⊂ C[[t]] is continuous. Such algebras were first introduced in [13] when considering algebras with one generator, and studied, in a special case, in [23]. For a partial bibliography of their subsequent study and application see the references of [9] (note that the usage of the term Banach algebra of power series in [9] differs from that here), and also [2], [3], [11]. Indeed, an examination of their use in [11], under more general topological conditions than here, led the present author to the results of [14], [15], [16], [17].

The problem of diffraction of a Kelvin wave by a narrow opening in a barrier separating two semi-infinite sheets of water is formulated as an integral equation. An approximate solution is obtained on the hypothesis that the gap is small compared with the length of the incident wave. The scattered disturbance comprises both Kelvin and Poincaré waves, and asymptotic formulate for their amplitudes are obtained. The results, which are relevant to tidal diffraction by a strait, reveal that the phase shift in the diffracted Kelvin wave is much larger than the corresponding shift induced by either a narrow channel or an inland sea opening into a semi-infinite ocean.

During his investigation of the possible non-Hopf kernels for finitely generated groups in [1], Dey proves that the minimum number of generators d(Gn) of the n-th direct power Gn of a non-trival finite group G tends to infinity with n. This has prompted me to ask the question: what are the ways in which the sequence {d(Gn)} can tend to infinity? Let us call this the growth sequence for G; it is evidently monotone non-decreasing, and is at least logarithmic (Theorem 2.1). This paper is devoted to a proof that, broadly speaking, there are two different types of behaviour. If G has non-trivial abelian images (the imperfect case, § 3), then the growth sequence of G is eventually an arithmetic progression with common difference d(G/G'). In special cases (Theorem 5.2) the initial behaviour can be quite nasty. Our arguments in § 3 are totally elementary. If G has only trivial abelian images (the perfect case,§ 4), then the growth sequence of G is eventually bounded above by a sequence that grows logarithmically. It is a simple consequence of this fact that there are arbitrarily long blocks of positive integers on which the growth sequence takes constant values. This is a characteristic property of perfect groups, and indeed it was this feature in the growth sequences of large alternating groups (which I found by using ad hoc permutational arguments) that attracted me to the problem in the first place. The discussion of the perfect case rests on the lovely paper of Hall [2], which was brought to my notice by M. D. Atkinson.

If x and y are elements in the group G, then we denote their commutator by x o y = x-1y-1 = x-1xy; and x o G is the set of all commutators x o g with g ∈ G. A G-commutator sequence is a series of elements ci ∈ G with c1 + 1 ∈ ci O G. Slightly generalizing well known results one proves that the hypercenter of the group G is exacly the set of all elements h ∈ G with the property: every G-commutator sequence, containing h, contains 1 [Proposition 1.1].

The purpose of this paper is to give a more precise form of Theorem 1 of [2], which gives a structure theorem for subgroups of HNN groups; we prove the following.

Let H be a subgroup of the HNN group <A, xi;xiU-ixi-1 = Ui>. Then H is an HNN group whose base is a tree product of groups H ∪ wAw-1 where w runs over a set of double coset representatives of (H,A); the amalgamated and associated subgroups are all of the form H ∊ vUiv-l for some v. We can be more precise about which subgroups occur and about the tree product. We will also obtain stronger forms of other results in [1] and [2].

The isomorphism problem for finite groups of odd order and nilpotency class 2 with cyclic centre will be solved using some results of Brady [1], [2]. Since a finite nilpotent group is the direct product of its Sylow subgroups, we only need to consider finite q-groups where q is a prime. It has been shown in [1] and [2] that a finite q-group of nilpotency class 2 with cyclic centre is a central product either of two-generator subgroups with cyclic centre or of two-generator subgroups with cyclic centre and a cyclic subgroup, and that the q-groups of class 2 on two generators with cyclic centre comprise the following list: , and if q = 2 we have as well .

Freudenthal [5, 7] defined a compactification of a rim-compact space, that is, a space having a base of open sets with compact boundary. The additional points are called ends and Freudenthal showed that a connected locally compact non-compact group having a countable base has one or two ends. Later, Freudenthal [8], Zippin [16], and Iwasawa [11] showed that a connected locally compact group has two ends if and only if it is the direct product of a compact group and the reals.

The main result of this paper is the theorem in the title. Only special cases of it seem to be known so far. As an application, we obtain a result on the unique extension of Galois connexions. As a matter of fact, it is only by the use of Galois connexions that we obtain the main result, in its present generality.

1. General Idea. In this paper formal operators and bracketings devices are essentially the same; so are well formed formulae and correct bracketings.

A well known theorem of symbolic logic (Post languages, etc.) and universal algebra characterizes well formed formulae among (linear) strings of symbols for variables and operators in terms of a system of numerical inequalities, one of them an equality.

LetP1, P2…, pN be N points in the unit s-dimensional closed square Q = [0, 1]s. For any measurable set S ⊆ Q, we define δ(S), the discrepancy of S, by , where V(S) is the s-dimensional volume of S, and n(S), is the number of indices i for which pi∈S. Let , where the supermum is taken over all s-balls B ∈ Q, and , the supermum in this case being taken over all convex sets C ∈ Q. Clearly Dc ≧ Dk. In this paper we establish Theorem , where φ1is a constant depending only on s.

Hawkes showed in [10] that classes of metanilpotent groups which are both formations and Fitting classes are saturated and subgroup closed; more, he characterized all such classes as those local formations with a local definition consisting of saturated formations (of nilpotent groups). In [3] we showed that those “Fitting formations” which are subgroup closed are also saturated, without restriction on nilpotent length; indeed such classes are, roughly speaking, recursively definable as local formations using a local definition consisting of such classes. It is natural to ask how these hypotheses may be weakened yet still produce the same classes of groups. Already in [10] Hawkes showed that Fitting formations need be neither subgroup closed nor saturated; and in [3] we showed that a saturated Fitting formation need not be subgroup closed (though a Fitting formation of groups of nilpotent length three is saturated if and only if it is subgroup closed).

This paper is an outgrowth of my old battle with the open sentence problem for the theory of finite groups. The unsolvability of the word problem for groups (cf. [1] and [4]) entails the undecidability of the open sentence problem for the elementary theory of groups and thus strengthens the original undecidability result for this theory (cf. [7]). The fact that the elementary theory of finite groups is also undecidable (cf. [2] and [6]) therefore justifies my interest in the open sentence problem for that theory. This paper contains a construction of groups that might lead to a negative solution.

In her book on varieties of groups, Hanna Neumann posed the following problem [13, p. 166]: “

Can a variety other than D contain an infinite number of non-isomorphic non-abelian finite simple groups?”

The answer to this question does not seem to be known at present. However, in [7], Heineken and Neumann described an algorithm for determining whether or not there are any non-abelian finite simple groups satisfying a given law. They also outlined a way in which their algorithm could be used to show that “only finitely many of the known non-abelian finite simple groups can satisfy a given non-trivial law”; in this paper, we shall follow their suggestions, and prove the

THEOREM. Let g be a set of mutually non-isomorphic non-abelian finite simple groups, each of which is either an alternating group or a group of Lie type, and let g generate a proper subvariety of D. Then y is finite.

This paper is concerned with version of Bernstein's inequality for Hausdroff locally compact Abelian groups. The ideas used are suggested by Exercise 12, p. 17 of Katznelson's book [4].

It was shown by Baer in [1] that every soluble group satisfying Min-n, the minimal condition for normal subgroups, is a torsion group. Examples of non-soluble locally soluble groups satisfying Min-n have been known for some time (see McLain [2]), and these examples too are periodic. This raises the question whether all locally soluble groups with Min-n are torsion groups. We prove here that this is not the case, by establishing the existence of non-trivial locally soluble torsion-free groups satisfying Min-n. Rather than exhibiting one such group G, we give a general method for constructing examples; the reader will then be able to see that a variety of additional conditions may be imposed on G. It will follow, for instance, that G may be a Hopf group whose normal subgroups are linearly ordered by inclusion and are all complemented in G; further, that the countable groups G with these properties fall into exactly isomorphism classes. Again, there are exactly isomorphism classes of countable groups G which have hypercentral nonnilpotent Hirsch-Plotkin radical, and which at the same time are isomorphic to all their non-trivial homomorphic images.