For cardinal numbers λ, K, ∑ a (λ, K)-family is a family of sets such that || = and |A| = K for every A ε , and a (λ, K, ∑)-family is a (λ,K)-family such that |∪| = ∑. Two sets A, B are said to be almost disjoint if

and an almost disjoint family of sets is a family whose members are pairwise almost disjoint. A representing set of a family is a set X ⊆ ∪ such that X ∩ A = ⊘ for each A ε . If is a family of sets and |∪| = ∑, then we write εADR() to signify that is an almost disjoint family of ∑-sized representing sets of . Also, we define a cardinal number

Let K(k) and E(k) denote respectively the complete elliptic integrals of the first and second kind with modulus k, as defined by Byrd and Friedman ([5] 110·06 and 110·07), and let k′ = √(1 − k2), the complementary modulus.

Burnside[1] considered possible restrictions on the derived group G′ of p-group G and showed that if G′ is non-Abelian, the centre Z(G′) of G′ is not cyclic. This implies that |G′: G″| ≥ p3. Many other restrictions on G′ are to be found in Hall's famous paper [2], but in 1954 Hall proved that if p is odd and |G′: G″| = p3, then |G′| ≤ p. So far as I know, no proof of this is to be found in the literature, but it follows from the lemma below. Our concern here is with the case p = 2, and we shall prove the following.

It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutations

generate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n (< ∞) symbols has rank 3, being generated by the two generators of Sn, together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Singn, the semigroup of all singular self-maps of {1, …, n}, is harder to determine: in Section 2 it is shown to be ½n(n − 1) (for n ≽ 3). The semigroup Singn it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Singn as the cardinality of the smallest possible set P of idempotents for which <F> = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.

Given a group G, a subgroup H and an element a ∈ G, we define Γ(G, H, a) to be the graph on the set of left-cosets of H in G, where two left-cosets g1H and g2H are adjacent whenever . We will consider this construction only in the case that

The first of these conditions assures that adjacency is a symmetric relation (i.e. that Γ(G, H, a) is an undirected graph) and the second assures that Γ(G, H, a) is connected.

If F is a free profinite group, it is well known that the closed subgroups of F need not be free profinite; however, if p is a prime number, every closed subgroup of a free pro-p-group is free pio-p (cf. [2, 8, 7]). In this paper we show that there is an analogous contrast regarding the closed subgroups of free products in the category of profinite groups, and the closed subgroups of free products in the category of pro-p-groups, at least for (topologically) finitely generated subgroups.

By exploiting the well known spin representations of the orthogonal groups O(l), Morris [12] was able to give a unified construction of some of the projective representations of Weyl groups W(Φ) which had previously only been available by ad hoc means [5]. The principal purpose of the present paper is to give a corresponding construction for projective representations of the rotation subgroups W+(Φ) of Weyl groups. Thus we construct non-trivial central extensions of W+(Φ) via the well-known double coverings of the rotation groups SO(l). This adaptation allows us to give a unified way of obtaining the basic projective representations of W+(Φ) from those of W(Φ) determined in [12]. Hence our work is a development of the recent work of Morris, and is an extension of Schur's work on the alternative groups [15].

The aim of this paper is to prove some results involving Weyl groups which are useful in an application to give a combinatorial construction of representations of Weyl groups which generalizes the well-known Young tableau method for symmetric groups. A preliminary version of these applications has been given in [12]. As these results seem to be of independent interest they are gathered together in the present paper.

Let R denote a commutative Noetherian ring with an identity element. Hilbert's basis theorem states that the polynomial ring in a finite number of indeterminates over R is also Noetherian. (See Northcott ], theorem 8, p. 26; Zariski and Samuel [4], theorem 1, p. 201). Hilbert's original theorem in [2] is stated for the case when R is a field or the ring of integers. The standard proofs of this fundamental theorem are essentially of a direct type. The analogue of Hilbert's basis theorem in the ring of formal power series in a finite number of indeterminates over R is also true (Chevalley [1]; see also Northcott [3], theorem 3, p. 89; Zariski and Samuel [5], theorem 4, p. 138). In the present note we bring together concise proofs of Hilbert's and Chevalley's theorems by using a single indirect (i.e. reductio ad absurdum) argument. By induction it is sufficient to consider the case of a polynomial ring S and a formal power series ring T in a single indeterminate X over R.

Let R be a commutative ring with identity and let q be an ideal in R. For each n ≽ 2, let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(R, q) be the normal subgroup of En(R) generated by the q-elementary matrices. We put SLn(R, q) = Ker(SLn(R)→SLn(R/q)), the principal congruence subgroup of GLn(R) of level q. (By definition En(R, R) = En(R) and SLn(R, R) = SLn(R).)

This paper is a continuation of [1], where we introduced the concept and investigated the basic properties of Fitting functors for finite solvable groups. We recall that a map f which assigns to each finite solvable group G a non-empty set f(G) of subgroups of G is called a Fitting functor if the following property is satisfied: whenever α: G → H is a monomorphism with α(G) ⊴ H, then

The most prominent examples of conjugate Fitting functors are provided by injectors of Fitting classes. However, Fitting functors f in general do not behave as nicely as injectors. For instance, f(G) need not consist of pronormal subgroups of a group G (cf. [1], 4·3).

We prove the existence of unique limits and establish inequalities for matrix generalizations of the arithmetic–geometric mean of Lagrange and Gauss. For example, for a matrix A = (aij) with positive elements aij, define (contrary to custom) A½ elementwise by [A½]ij = (aij)½. Let A(0) and B(0) be d × d matrices (1 < d < ∞) with all elements positive real numbers. Let A(n + 1) = (A(n) + B(n))/2 and B(n + 1 ) = (d−1A(n)B(n))½. Then all elements of A(n) and B(n) approach a common positive limit L. When A(0) and B(0) are both row-stochastic or both column-stochastic, dL is less than or equal to the arithmetic average of the spectral radii of A(0) and B(0).

The notion of a reduction of an ideal I of a noetherian ring R and the related notions of the integral closure I* of I in R and the integral dependence of an element x of R on I have proved useful in many situations in commutative algebra. The purpose of this paper is to extend these notions to modules. To motivate the definitions which follow, it will be useful to consider these notions for ideals before extending them to modules.

Akbulut and King [3, 4] have obtained several interesting results investigating the effects on homology of real algebraic varieties of the blowing-up construction. Here we apply this technique to study the behaviour of homotopy classes.

Let and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

Recently, the present author together with M. Crampin proved a structure theorem for a certain subclass of geometric objects known as almost tangent structures (Crampin and Thompson [8]). As the name suggests, an almost tangent structure is obtained by abstracting one of the tangent bundle's most important geometrical ingredients, namely its canonical 1–1 tensor, and using it to define a certain class of G-structures. Roughly speaking, the structure theorem referred to above may be paraphrased by saying that, if an almost tangent structure is integrable as a G-structure and satisfies some natural global hypotheses, then it is essentially the tangent bundle of some differentiable manifold. (I shall have a further remark to make about the conclusion of that theorem at the end of Section 2.)

In this paper we list the elliptic curves defined over Q(√− 3) with good reduction away from the prime dividing 3. As in [8] and [9] a discriminant estimate is used to show that such a curve must have a subgroup of order 3 defined over Q(√−3).

It is shown that an abelian group admits a non-discrete locally compact group topology if and only if it has a subgroup algebraically isomorphic to the group of p-adic integers or to an infinite product of non-trivial finite cyclic groups. It is also proved that an abelian group admits a non-totally-disconnected locally compact group topology if and only if it has a subgroup algebraically isomorphic to the group of real numbers. Further, if an abelian group admits one non-totally-disconnected locally compact group topology then it admits a continuum of such topologies, no two of which yield topologically isomorphic topological groups.

The concept of arcbody was introduced in [4] as a generalization of prime tangles used by Lickorish[3] and Bleiler[l]. The pair (A, l), where A is a compact 3-manifold with boundary ∂A, and l is a 1-submanifold properly embedded in A, is an arcbody if:

(i) the inclusion ∂ A – l ↪ A – l is monic, i.e. A – l has incompressible boundary;

(ii) no component of (A, l) is homeomorphic to (D2, 0) × I or (D2, 0) × S1; and

(iii) every sphere of ∂A intersects l.

We observe that if is a compatible totally bounded quasi-uniformity on a T0-space (X,), then the bicompletion of (X, ) is a strongly sober, locally quasicompact space. It follows that the b-closure S of (X, ) in is homeomorphic to the sobrification of the space (X, ). We prove that S is equal to if and only if (X, ) is a core-compact space in which every ultrafilter has an irreducible convergence set and is the coarsest quasi-uniformity compatible with . If is the Pervin quasi-uniformity on X, then S is equal to if and only if X is hereditarily quasicompact, or equivalently, is the Pervin quasi-uniformity on .