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A new proof is given of the connecting homomorphism.

Let $N$ be a nilpotent group with torsion subgroup $TN$, and let $\alpha: TN\rightarrow \tilde T$ be a surjective homomorphism such that $\operatorname{ker}\alpha$ is normal in $N$. Then $\alpha$ determines a nilpotent group $\tilde N$ such that $T\tilde N=\tilde T$ and a function $\alpha_*$ from the Mislin genus of $N$ to that of $\tilde N$ if $N$ (and hence $\tilde N$) is finitely generated. The association $\alpha\mapsto\alpha_*$ satisfies the usual functorial conditions. Moreover $[N,N]$ is finite if and only if $[\tilde N,\tilde N]$ is finite and $\alpha_*$ is then a homomorphism of abelian groups. If $\tilde N$ belongs to the special class studied by Casacuberta and Hilton (Comm. in Alg. 19(7) (1991), 2051--2069), then $\alpha_*$ is surjective. The construction $\alpha_*$ thus enables us to prove that the genus of $N$ is non-trivial in many cases in which $N$ itself is not in the special class; and to establish non-cancellation phenomena relating to such groups $N$.

Proceedings of the London Mathematical Society, s3-36 (2), ISSN:0024-6115, ISSN:1460-244X

I had the good fortune to work closely with Alan Turing and to know him well for the last 12 years of his short life. It is a rare experience to meet an authentic genius. Those of us privileged to inhabit the world of scholarship are familiar with the intellectual stimulation furnished by talented colleagues. We can admire the ideas they share with us and are usually able to understand their source; we may even often believe that we ourselves could have created such concepts and originated such thoughts. However, the experience of sharing the intellectual life of a genius is entirely different; one realizes that one is in the presence of an intelligence, a sensitivity of such profundity and originality that one is filled with wonder and excitement. Alan Turing was such a genius, and those, like myself, who had the astonishing and unexpected opportunity created by the strange exigencies of the Second World War to be able to count Turing as colleague and friend will never forget that experience, nor can we ever lose its immense benefit to us. Before the war, in 1935–36, Turing had done fundamental work in mathematical logic and had invented a concept that has come to be known as the ‘universal Turing machine’ (see Chapter 6). His purpose was to make precise the notion of a computable mathematical function, but he had in fact provided a blueprint for the most basic principles of computer design and for the foundations of computer science. I joined the distinguished team of mathematicians and first-class chess players working on the Enigma code in January 1942. Alan Turing was the acknowledged leading light of that team. However, I must emphasize that we were a team—this was no one-man show! Indeed, Turing’s contribution was somewhat different from that of the rest of the team, being more concerned with improving our methods, especially the machines we used to help us, and less concerned with our daily output of deciphered messages. It was due to the efforts of Turing and the entire team that Churchill was able to describe our work as ‘my secret weapon’.

SummaryThis paper describes how the authors used a set of magnetic toys to discover analogues in 3 dimensions of well known theorems about binomial coefficients. In particular, they looked at the Star of David theorem involving the six nearest neighbors to a binomial coefficient . If one labels the vertices of the bounding hexagon with the numbers 1, 2, 3, 4, 5, 6, consecutively (in either direction), then the product of the coefficients with even labels is the same as the product as the coefficients with odd labels. Furthermore the two figures formed by connecting the odd and even vertices are both equilateral triangles arranged so that a rotation of ±60° exchanges the triangles. There is a generalized Star of David theorem concerning a semi-regular hexagon with similar results. The paper describes analogous results for trinomial coefficients involving, sometimes but not always, tetrahedra instead of triangles.

Dedicated to the memory of Russell Towle, a remarkable man who contributed so much to geometry and to other aspects of the quality of life.We introduce an expanded notation where r + s = n, for the binomial coefficient , and then use this expanded notation to develop theorems involving 8 binomial coefficients, analogous to the Star of David Theorem, which. in its original form, involved the 6 neighbours of a given binomial coefficient in the Pascal Triangle (see Section 3), that appeared in [1,2,3,4,5,6,7,8,9].

A general study is undertaken of product-wedge-diagonal (= PWD ) structures on a space. In part this concept may be viewed as arising from G.W. Whitehead's fat-wedge characterization of Lusternik–Schnirelmann category. From another viewpoint PWD -structures occupy a distinguished position among those structures that provide data allowing Hopf invariants to be defined. Indeed the Hopf invariant associated with a PWD -structure is a crucial component of the structure. Our overall theme addresses the basic question of existence of compatible structures on X and Y with regard to a map X → Y . A principal result of the paper uses Hopf invariants to formulate a Berstein–Hilton type result when the space involved is a double mapping cylinder (or homotopy pushout). A decomposition formula for the Hopf invariant (extending previous work of Marcum) is provided in case the space is a topological join U * V that has PWD -structure defined canonically via the join structure in terms of diagonal maps on U and V .

In [1, Chapter 3, Section 2], we collected together results we had previously obtained relating to the question of which positive integers m were Lucasian, that is, factors of some Lucas number Ln. We pointed out that the behaviors of odd and even numbers m were quite different. Thus, for example, 2 and 4 are both Lucasian but 8 is not; for the sequence of residue classes mod 8 of the Lucas numbers, n ⩾ 0, readsand thus does not contain the residue class 0*. On the other hand, it is a striking fact that, if the odd number s is Lucasian, then so are all of its positive powers.

In a system of interlocking sequences, the assumption that three out of the four sequences are exact does not guarantee the exactness of the fourth. In 1967, Hilton proved that, with the additional condition that it is differential at the crossing points, the fourth sequence is also exact. In this paper, we trace such a diagram and analyze the relation between the kernels and the images, in the case that the fourth sequence is not necessarily exact. Regarding the exactness of the fourth sequence, we remark that the exactness of the other three sequences does guarantee the exactness of the fourth at noncrossing points. As to a crossing pointp, we need the extra criterion that the fourth sequence is differential. One notices that the condition, for the fourth sequence, that kernel⊇image atpturns out to be equivalent to the “opposite” condition kernel⊆image. Next, for the kernel and the image atpof the fourth sequence, even though they may not coincide, they are not far different—they always have the same cardinality as sets, and become isomorphic after taking quotients by a subgroup which is common to both. We demonstrate these phenomena with an example.

We describe in detail Serre's application of spectral sequence theory to the study of the relations between the homology of total space, base space and fibre in a Serre fibration; and we apply the results to establish that a 1-connected space X has homology groups (in positive dimension) in a Serre class C if and only if its homotopy groups are in C . We include in this paper some personal reflections on the contact the author had with Serre during the decade of the 1950's when Serre's revolutionary work in homotopy theory was completely changing the face of algebraic topology.

We give a sustained example, drawn largely from earlier publications, of how we may freely pursue a line of mathematical enquiry if we are not constrained, unnaturally, to confine ourselves to a single mathematical subdiscipline; and we draw conclusions from the study of this example which are relevant at many levels of mathematical instruction. We also include the statement and proof of a new result (Theorem 4.1) in linear algebra which is obviously fundamental to the geometrical investigation which constitutes the leit-motif of the paper.

A rational number is said to be expressed in Egyptian form (or Egyptian presentation) if it is written as a sum of fractions where each fraction has the numerator 1 and all of the denominators are distinct. The individual terms of die sum are often referred to as unit fractions. The problem of finding such expressions, and many related questions, has a long history, apparently beginning with Ahmes in the Rhind Papyrus. It is for this reason that we talk of ‘Egyptian’ form.

(1999). Relating Geometry and Algebra in the Pascal Triangle, Hexagon, Tetrahedron, and Cuboctahedron Part II: Geometry and Algebra in Higher Dimensions: Identifying the Pascal Cuboctahedron. The College Mathematics Journal: Vol. 30, No. 4, pp. 279-292.

We prove that there is a bijection between the Mislin genus of a circle bundle over a certain nilpotent base space M M , which is constructed from a nilpotent group N N of a certain specified type, and the Mislin genus of N N itself.

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts , only one group were problem-solving experts . Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.

(1997). The Faces of the Tri-Hexaflexagon. Mathematics Magazine: Vol. 70, No. 4, pp. 243-251.

We will discuss in this article a fractal-like structure made from a flat piece of paper. What will motivate most people to want to make the structure is that it is pretty. In fact, the exercise has two rather obvious uses. First, greeting card companies may want to use the idea to manufacture interesting 3-dimensional cards which fit conveniently into envelopes; and, second, teachers may wish to teach students how to make it (and this process will also involve teaching them some beautiful mathematics). The structure itself is a model for a stage in a self-similarity process leading to a fractal. Building the model involves scoring a flat piece of paper in a prescribed manner, cutting along some of the score lines, and then folding some lines as ‘mountain’ folds and others as ‘valley’ folds.

A nilpotent groupGis fgp ifGp, is finitely generated (fg) as ap-local group for all primesp; it is fg-like if there exists a nilpotent fg groupHsuch thatGp≃Hpfor all primesp. The fgp nilpotent groups form a (generalized) Serre class; the fg-like nilpotent groups do not. However, for abelian groups, a subgroup of an fg-like group is fg-like, and an extension of an fg-like group by an fg-like group is fg-like. These properties persist for nilpotent groups with finite commutator subgroup, but fail in general.

and which is not primarily erected for this purpose. The applications bear witness to the power of mathematics, but are not its real motivation. The springs of 686 [October MATHEMATICS: QUESTIONS AND ANSWERS This content downloaded from 207.46.13.120 on Wed, 14 Sep 2016 04:15:20 UTC All use subject to http://about.jstor.org/terms

The Mislin genus $\Cal G(N)$ of a finitely generated nilpotent group $N$ with finite commutator subgroup admits an abelian group structure. If $N$ satisfies some additional conditions ---we say that $N$ belongs to $\Cal N_1$--- we know exactly the structure of $\Cal G(N)$. Considering a direct product $ N_1 \times \cdots \times N_k$ of groups in $\Cal N_1$ takes us virtually always out of $\Cal N_1$. We here calculate the Mislin genus of such a direct product.

In their stimulating paper [1], to which we here make a friendly and constructive response the authors, Branko Grunbaum and Geoffrey Shephard, introduce the interesting geometric concept of polyhedral set, generalizing the familiar notion of polyhedron, but confining themselves, for their present purposes, to subsets of 1R3. They discuss dissections of such sets, especially relatively open convex dissections of bounded polyhedral sets and show, by easily accessible arguments, the nice properties of the Euler characteristic X relative to such dissections. They are thereby led to a formula for X(P), namely, Theorem 4 of [1]

In three earlier articles we drew attention to an astonishing error which appears in several texts in the United States, even some of otherwise very high quality. The error relates to a fallacious procedure for the calculation of the Euler characteristic of a polyhedron, producing, in particular, the extraordinary statement that the Euler characteristic of a torus is 2.

Let N be a P-local nilpotent group. We say that N is finitely generated (fg) as P-local group if there is a finite subset S of N such that the smallest P-local subgroup of N containing S is N itself; we could also say that S generates N as P-local group and write N=〈S〉 p. If G is nilpotent, we say that G is finitely generated at every prime (fgp) if Gp is fg as p-local group for all primes p. Such groups share with fg nilpotent groups many important properties (e.g., they are Hopfian). We show that they satisfy the axioms for a Serre class, as extended by Hilton and Roitberg to nilpotent groups.

Disclaimer This article is full of generalizations, many about student attitudes. Such generalizations, unlike generalizations in mathematics, are not invalidated by the existence of counterexamples. All of us know the joy of teaching those exceptional undergraduates who really want to learn and are stimulated by the beauty and power of the mathematics they are learning. It is these students who give us the conviction that we are, as teachers, doing something thoroughly worthwhile, despite our many failures. 1. INTRODUCTION. I have been distressed for many years by the defects in our methods of evaluating students-at all levels from 3rd grade upwards to graduate students-and have railed especially against the unfortunate effects of over-frequent testing and inappropriate tests, especially multiple-choice and standardized tests. I have not changed my point of view on these issues; but I have come to realize that the ubiquity of intrusive testing, contaminating the learning environment, distorting the curriculum, and undermining the fabric of the teacher-student relationship, springs from a deep malaise in our whole approach to education, and is not to be remedied simply by attempting to improve the quality and relevance of the tests administered and reducing their frequency. Such a superficial approach may be thought of as treating a symptom and leaving the cause of the disease untouched. In this article I would like to develop this thought, and then raise the question of whether any change-any beneficial change, that is-is possible. I will largely confine myself to undergraduate education-principally, of course, in mathematics. 2. THE ROOT CAUSE OF THE PROBLEM. I have often remarked that, in this country, there is a fundamental confusion of education with training. Thus, whereas drivers of lethal automobiles stand in evident need of first-class training, we provide 'driver education'; and whereas teachers stand in evident need of first-class education, we provide 'teacher training.' Indeed, I would now go further and say that, in general, our students-and their parents-only feel comfortable with the presence of a curricular item if it has an evident value for the students'