In a previous paper [1] we constructed a free resolution for a class of groups which include Fuchsian groups with compact orbit spaces [2, 3], infinite polyhedral groups, plane crystallographic groups p2, p3, p4 and p6 and Dyck's groups [4], and used this resolution for computation of the integral homology and cohomology of these groups. Lyndon [5] determined the cohomology of groups with a single defining relation. The plane crystallographic groups p1 and pg and Artin's braid group B3 are among these groups. In this paper we have constructed free resolutions for certain classes of groups–resolutions which are particularly suitable for direct computation of the homology and the cohomology of these groups for any coefficient module. These classes of groups include the plane crystallographic groups pm, cm and pgg. We have computed the integral homology and cohomology from each of the free resolutions obtained.

1. If p and q be both commensurable, and if , then, if n be prime to r, both the roots must be commensurable.

For, since n is prime to r, we can find two integers λ and μ such that λn + μr = 1.

Dieudonné (4) has constructed an example of a Banach space X and a complete Boolean algebra of projections on X such that has uniform multiplicity two, but for no choice of x1, x2 in X and non-zero E in is EX the direct sum of the cyclic subspaces clm {Ex1:E∈} and clm {Ex2:E∈}. Tzafriri observed that it could be deduced from Corollary 4 (9, p. 221) that the commutant ′ of is equal to A(), the algebra of operators generated by in the uniform operator topology. A study of (3) suggested the direct proof of the second property given in this note. From this there follows a simple proof that has the first property.

In this note it is shown that Theorem 2.8 of [2] is erroneous. Throughout this note nea-ring means a right near-ring.

We shall denote by ω the natural partial order on the idempotents E = E(S) of a regular semigroup S, so that in E,

A partially ordered semigroup S(≦) is called naturally partially ordered [9] if the imposed partial order ≦ extends ω in the sense that

No assumption is made about the reverse implication.

Let A be a matrix over a field Φ partitioned as follows

where A11 is n×n and A22 is m×m. The objective of the present paper is to give further results on the problems mentioned in Section 1 of (3). Concretely we shall consider the following question: “we prescribe the characteristic polynomial f(λ) = λn+m−c1λn+m−1 + … of A and the principal blocks A11, A22. Find a necessary and sufficient condition for the existence of A satisfying these prescribed conditions”.

In the present note certain known theorems on the latent roots of matrices are deduced from the fundamental theorem that a matrix A can be expressed in the form PQP-1, where P is non-singular and Q has zero elements everywhere to the left of the principal diagonal, and the latent roots of A in the diagonal. [The presence or absence of non-zero elements to the right of the diagonal is known to depend on the nature of the “elementary divisors” of the “characteristic determinant” of A, but in what follows these will not concern us.]

It is well known that the properties of the orthocentre and of the nine-point circle of a triangle may be most symmetrically stated when the triangle and its orthocentre are looked upon as the vertices of a four-point, the opposite sides of which intersect at right angles. This point of view leads naturally to a generalisation of the ninepoint circle, by consideration of any four-point in place of the orthic four-point—a generalisation which was first given in detail by Beltrami in the year 1863; though the theorems involved had been previously stated by T. T. Wilkinson. A number of papers have since been written on the nine-point conic; but they have for the most part merely given Beltrami's results over again, and have generally been written in ignorance of his work. In this paper I propose giving the properties of the nine-point conic from a different point of view, associating them with the triangle instead of the four-point. There are certain advantages belonging to each point of view. If, for instance, we consider a triangle ABC with its orthocentre H as an orthic four-point, any proof that shows that the nine-point circle touches the inscribed (or an escribed) circle of the triangle ABC, will, in general, also show that it touches the inscribed (and escribed) circles of the triangles HCB, CHA and BAH. On the other hand, as the nine-point conic circumscribes the diagonal triangle of the four-point, if the four-point is given, the nine-point conic is definitely determined; whereas, if the triangle be considered, as the fourth vertex of the four-point may be taken arbitrarily, a number of nine-point conics are obtained, touching the same inscribed conic.

Let l be the length of the rectangle and b its breadth, while s is the side of the square equal to it in area; s is found, of course, by taking l: s = s : b. A number of different cases arise.

One can define “Laurent” and “Hankel” operators relative to groups more general than the circle group . We do that here, derive some of their properties, and compute their spectra, using a concrete realization of a crossed product C*-algebra by the two-element group ℤ2.

The major problem with which this paper is concerned is determining criteria that allow one to decide whether the subsemigroup generated by a subset B of a group G is all of G. Motivations for considering this problem arise from at least two sources.

If the group of inner automorphisms of a semigroup S of transformations of a finite n-element set contains an isomorphic copy of the alternating group Altn, then S is an Sn-normal semigroup and all the automorphisms of S are inner.

The gigantic task which Mr Cantor has undertaken in writing a history of mathematics down to the year 1759 is approaching its accomplishment, the first two of the three parts forming the third and concluding volume being now published. How great is the debt of gratitude that the mathematical public owe to him for the erudition and thoroughness he has brought to bear on the work, only those can guess who have attempted to follow out some line of historical investigation. His history is totally different from a catalogue of authors and their works; it enables us to trace clearly the lines of development of mathematical knowledge and shows with wonderful skill not only the contributions made by the creators of the science but the conditions and materials that made these possible. The fact that the greatest mathematicians recognise that they obtained their position because “they stood on the shoulders of great men” does not in the least detract from their merits but is an abiding augury for future developments.