§ 1. Introductory. The formula to be established is

where m is a positive integer,

and the constants are such that the integral converges.

The norm K(G) of a group G is the subgroup of elements of G which normalize every subgroup of G. Under the name kern this subgroup was introduced by Baer [1]. The norm is Dedekindian in the sense that all its subgroups are normal. A theorem of Dedekind [5] describes the structure of such groups completely: if not abelian they are the direct product of a quaternion group of order eight and an abelian group with no element of order four. Baer [2] proves that a 2-group with non-abelian norm is equal to its norm.

In recent years various authors have studied integral operators involving confluent hypergeometric functions Mκ,μ and Wκ,μ. Using the method devised by Fox [2], Saxena [5] obtained the inverse of an integral operator with kernel (xt) μ− ½e− ½xtWκ,μ (xt). Singh [6] derived the solution of an integral equation of convolution type with kernel

If f is an entire function in the complex plane such that

where 0 ≤ α < 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).

There is no example known of a non-nilpotent, torsion-free group which has all of its subgroups subnormal. It was proved in [3] that a torsion-free solvable group with all of its proper subgroups subnormal and nilpotent is itself nilpotent, but that seems to be the only published result in this area which is concerned specifically with torsion-free groups. Possibly the extra hypothesis that the group be hypercentral is sufficient to ensure nilpotency, though this is certainly not the case for groups with torsion, as was shown in [7]. The groups exhibited in that paper were seen to have hypercentral length ω + 1, and we know from [8] that further restricting the hypercentral length can lead to some positive results. Here we shall prove the following theorem.

In this paper, a problem of B. H. Neumann and Hanna Neumann [7] about the finite embeddability of an embeddable finite amalgam is discussed. After proving a “reduction theorem” for a finite amalgam to have a finite embedding, we examine some known embeddable amalgams (cf. [3]) as regards their embeddability in a finite group. Since the existence of the generalised free product and the embeddability of an amalgam are synonymous terms, Theorem 3.1 generalises a result in [4]. A sufficient condition for an amalgam of type S to have a finite embedding is also given.

We consider a volume of material, divided into two regions 1 and 2. each of density ρ, by a moving surface S. On S a change of phase occurs, at a definite temperature (which we may take to be zero) and with absorption or liberation of a latent heat L per unit mass. If θl, kl, K1 are the temperature, thermal conductivity and diffusivity of phase 1, and θ2, k2, K2 corresponding quantities for phase 2, the surface S is the isothermal

and the boundary condition on this surface is

Subscript letters denote partial differentiation.

Throughout this paper R will be an associative ring with unity and all R-modules are unitary. The right (resp. left) annihilator in R of a subset X of a module is denoted by r(X)(resp. I(X)). The Jacobson radical of R is denoted by J(R), the singular ideals are denoted by Z(RR) and Z(RR) and the socles by Soc(RR) and Soc(RR). For a module M, E(M) and PE(M) denote the injective and pure-injective envelopes of M, respectively. For a submodule A ⊆ M, the notation A ⊆⊕M will mean that A is a direct summand of M.

The concept of weak relative-injectivity of modules was introduced originally in [10], where it is shown that a semiperfect ring R is such that every cyclic right module is embeddable essentially in a projective right R-module if and only if R is right artinian and every indecomposable projective right R-module is uniform and weakly R-injective. We show that in the above characterization the requirement that indecomposable projective right R-modules be uniform is superfluous (Theorem 1.11). In this paper we further the study of weak relative-injectivity by considering the class of rings for which every right module is weakly injective relative to every finitely generated right module. We refer to such rings as right weakly-semisimple rings. The class of right weakly-semisimple rings includes properly all semisimple rings and is a subclass of the class of right QI-rings. A ring R is said to be a right QI-ring if every quasi-injective right R-module is injective. QI-rings have been studied in [2], [3], [4], [6], [7], [8], [11], among others.

In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in λ colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed λ [1, 4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another.

Let {an} be a sequence of non-negative real numbers. Suppose that

Then M1,n is the arithmetic mean, MO,n the geometric mean, and Mr,n the generalized mean of order r, of a1, a2, …, an. By a result of Everitt [1] and McLaughlin and Metcalf [5], {n(Mr,n–Ms,n)}, where r ≧ l ≧ s, is a monotonic increasing sequence. It follows that this sequence tends to a finite or an infinite limit as n → ∞. Everitt [2, 3] found a necessary and sufficient condition for the finiteness of this limit in the cases r, s = 1, 0 and r ≧ 1 > s > 0. His results are included in the following theorem.

Let k be an even integer greater than or equal to 12 and f an nonzero cusp form of weight k on SL(2, Z). We assume, further, that f is an eigenfunction for all Hecke-Operators and has the Fourier expansion

For every Dirichlet character xmod Q we define

The main result of this series of papers is a theorem on the free product of groups (Theorem 1) which formed part of a doctoral thesis. This theorem has an immediate application to the word problem (Theorem 2). Usually the word problem refers to a finite system of generators and a finite number of defining relations, but in this context it is more natural to allow an infinite number of generators and defining relations. This (infinite) word problem is not solvable in general (Example 2).

Let G be a compact Lie group and X a G-CW complex. We are interested in the calculation of the Borel cohomology of X

where EG is a universal free G-space and we use on the right hand side cellular cohomology. For an introduction to G-CW complexes see Matumoto [4] and for a good exposition on Borel cohomology see for instance torn Dieck [2], We want to replace X with an ordinary CW complex Y in order to find an ordinary CW structure on the Borel construction EG ΧGY so we can use cellular chains to compute the Borel cohomology of X. For every compact Lie group one has an extension

where G0 is the identity component, so for our case G0 is isomorphic to the circle group . We are dealing with the case in which π0(G) is isomorphic to C2, the cyclic group of order

Let G be a finite group and p a prime number. About five years ago I. M. Isaacs and S. D. Smith [5] gave several character-theoretic characterizations of finite p-solvable groups with p-length 1. Indeed, they proved that if P is a Sylow p-subgroup of G then the next four conditions (l)–(4) are equivalent:

(1) G is p-solvable of p-length 1.

(2) Every irreducible complex representation in the principal p-block of G restricts irreducibly to NG(P).

(3) Every irreducible complex representation of degree prime to p in the principal p-block of G restricts irreducibly to NG(P).

(4) Every irreducible modular representation in the principal p-block of G restricts irreducibly to NG(P).

In this paper we calculate the localisation at the prime 3 of the integral cohomology ring of the Mathieu group M24, together with its mod-3 cohomology ring. The main results are

Theorem 1. The ring H*(M24, Z)(3)is the commutative graded Z(3)-algebra with generators

and relations v2 = 0 and βθ = 0. The Chern classes of the Todd representation in GL11F2

Let q be a power of a prime p, and let Sqbe the set of permutations of {0, 1,…, q – 1). As Sq is isomorphic to the group of permutations of Fq, the field of q elements, each element of 5, can be regarded as a polynomial over Fq. Various authors (e.g. [1], [2], [3]) have considered functions f(x) such that

f(x) ∈ Sq, and (f(x) + λ×) ∈ Sq

for some λ ∈ Fq when λ = 1, f(x) is a complete mapping polynomial ([3]).