In the course of classifying those finite groups F which have exactly five maximal subgroups, R. W. van der Waall [4] proved that one encounters the following situation. One class of such groups F is described by F = SP, where S = O2(F)∈Syl2(F), P ∈ Syl3(F), S/Φ(S) ≅ Z2 × Z2, P is cyclic and P operates via conjugation on 5 as a group of order 3, because in this case F/Φ(F) ≅ A4.

The following general problem is of interest. Let Λ be an irreducible algebraic variety of degree d, in projective n-space Pn, defined over a field k; and suppose that K is a finite extension of k with [K: k] prime to d. If Λ has a point defined over K, then does it necessarily have a point defined over k?

It has been studied in various instances by several authors: see, for example, Cassels [2], Coray [3, 4], Pfister [5], Bremner, Lewis, Morton [1]. Coray [3] shows that a quartic curve Λ over Q may possess points in extension fields of Q of every odd degree greater than one, but have no points in Q itself. Some further examples of this instance occur in the paper of Bremner, Lewis, Morton, with the additional property that the curve Λ also possesses points in every p-adic completion Qp of Q.

La fonction η de Dedekind est définie par

où , Im(z)>0. C'est une forme modulaire parabolique de poids 1/2. Si r est un entier, la puissance r–ième de η s'écrit;

où les coefficients pr(n) sone définis par l'identité

.

Let us consider the Cauchy problem

in QT= ℝ3 × [0, T], where f(x, t), ρ0(x) and v0(x) are given, while the density ρ(x, t), the velocity vector v(x, t)= (υ1(x, t), υ2(x, t), υ3(x, t)) and the pressure p(x, t) are unknowns. The viscosity coefficient μ is assumed to be nonnegative. In these equations, the pressure p is automatically determined (up to a function of t) by ρ and v, namely, by solving the equation

Thus we mention (ρ, v) when we talk about the solution of (1.1:μ).

If the set K of r+1 distinct integers k0, k1 …, kr has the property that the (r+1)r differences ki–kj (0≦i, j≦r, i≠j) are distinct modulo r2+r+1, K is called a perfect difference set modr2+r+1. The existence of perfect difference sets seems intuitively improbable, at any rate for large r, but in 1938 J. Singer [1] proved that, whenever r is a prime power, say r = pn, a perfect difference set mod p2n+pn+1 exists. Since the appearance of Singer's paper several authors have succeeded in showing that for many kinds of number r perfect difference sets mod r2+r+1 do not exist; but it remains an open question whether perfect difference sets exist only when r is a prime power (for a comprehensive survey see [2]).

All groups considered are finite. In recent years a number of generalizations of the classic Jordan-Hölder Theorem have been obtained (see [7], Theorem A.9.13): in a finite group G a one-to-one correspondence as in the Jordan-Holder Theorem can be defined preserving not only G-isomorphic chief factors but even their property of being Frattini or non-Frattini chief factors. In [2] and [13] a new direction of generalization is presented: the above correspondence can be defined in such a way that the corresponding non-Frattini chief factors have the same complement (supplement).

In this paper, we shall be concerned with the solution of triple integral equations of the type

where M−1 is the inverse Mellin transform, nis a positive integer, and — 1 < Res < 0. The use of these equations will be illustrated by their application to two well-known problems in the mathematical theory of elasticity and further applications will be reported later.

Let E be a finite set and let = (Al, …, A2) be a family of subsets of E. A subset T of E iscalled a transversal of if there exists a bijection Φ: T → {l, …, n} such that x ∈ Aψ(x) for all x ∈ T. If I ⊆ {1, …, n}, we shall, for brevity, write

(and similarly for families denoted by other letters). The cardinal of a set S will be denoted by |S|. If λ is a non-negative integer, we define λS as S or Ø according as λ > 0 or λ = 0.

Let μ be the Gaussian measure in ℂn given by dμ(z)=(2π)−n exp(−|z|2/2)dV, where dV is the ordinary Lebesgue measure in ℂn. The Segal-Bargmann space H2(μ) is the space of all entire functions on ℂn that belong to L2(μ)-the usual space of Gaussian square-integrable functions. Let P be the orthogonal projection from L2(μ) onto H2(μ). For a measurable function ϕ on ℂn, the multiplication operator Mϕ on L2(μ) is defined by Mϕh =ϕh. The Toeplitz operator Tϕ is defined on H2(μ)by

Throughout the paper we consider only finite groups.

J. C. Beidleman and H. Smith [3] have proposed the following question: “If G is a group and Ha subnormal subgroup of G containing Φ(G), the Frattini subgroup of G, such that H/Φ(G)is supersoluble, is H necessarily supersoluble? “In this paper, we give not only an affirmative answer to this question but also we see that the above result still holds if supersoluble is replaced by any saturated formation containing the class of all nilpotent groups.

Given a series we define , by the relations

The series is said to be summable (C, k), where k > - 1, to the sum s if

to be summable (C, - 1) to s if it converges to s and nan = o(l); to be absolutely summable (C, k), or summable | C, k, to s if it is summable (C, k) to s and

and to be strongly Cesàro summable to s with order k > 0 and index p or summable [C; k, p] to s, if

The notion of projective Banach module was defined by Helemskii in [1]—the paper which properly founded the homological theory of Banach algebras. The same author introduced the definition of the (relatively) flat Banach module in [2]. Recently M. C. White [3] modified both of those definitions, introducing so called C-projective and C-flat Banach modules.

In the paper [3] the following lemma was proved.

Lemma. Let a, b and c be positive integers such that a and be are relatively prime. Then there are infinitely many primes p in the arithmetic progression ax + b (x = 0,1,2,…) such that

Let S be a semigroup. A class of S-automata is called a hereditary pretorsion class (HPC) if it is closed under quotients, subautomata, coproducts (disjoint unions) and finite products. In this paper we present two characterizations of HPC. Specifically, we show that there is a bijective correspondence between the HPCs of S-automata, the right linear topologies on S′ and the idempotent preradicals r on the category of S-automata such that the set of automata {M|r(M) = M} is closed under subautomata and finite products.

After Dieudonné [3], many authors have considered locally convex topological vector spaces that admit a fundamental sequence of compact, precompact,… subsets. Their work has been essentially to identify them as strong duals of Fréchet-Montel spaces, under suitable additional assumptions (barrelledness or evaluability). However, it seems that these spaces have never been characterized without additional assumptions. That is the aim of the present paper.

If G1 and G2 are locally compact groups and the algebras Ll(G1) and Ll(G2) are isometrically isomorphic, then G1 and G2 are isomorphic (Wendel, 1952, [8]). There is evidence that the following generalization of Wendel's result is true.

If T is an algebra isomorphism of L1(G1) onto L1(G2) with ∥T∥ < √2, then G1, and G2 are isomorphic.

A distributive p-algebra is an algebra 〈L; ∨, ∧, *, 0, 1〉 for which 〈L, ∨, ∧, 0, 1〉 is a bounded distributive lattice and * is a unary operation on L such that a ∧ x = 0 if and only if x ≤ a* (i.e. a pseudocomplementation). A distributive double p-algebra is an algebra 〈L; ∨, ∧, *, +, 0, 1〉 in which the deletion of + gives a distributive p-algebra and the deletion of * gives a dual distributive p-algebra, that is a ∨ (x = 1 if and only if x ≥ a+.