Let G be a finite group and let S be a subgroup of G with

The number of distinct types of Abelian group of prime-power order pn is equal to the number of partitions of n. Let (ρ) = (ρ1, ρ2, …, ρr) be a partition of n and let (μ) = (μ1, μ2, …, μs) be a partition of m, with ρ1≧ρ2≧…≧ρr and μ1≧μ2≧…≧μs, ρi≧μi, r≧s, n>m. The number of subgroups of type μ in an Abelian p-group of type (ρ) is a function of the two partitions (μ) and p, and has been determined as a polynomial in p with integer coefficients by Yeh (1), Delsarte (2) and Kinosita (3). Their results differ in form but are equivalent.

In this paper we consider a function f(x) defined by

All quantities are taken to be real, it is assumed that R is a function of the variable x, b is a constant, N and G are functions of the variable t and all the functions are such that the integral (1) exists when x is large enough. We wish to find an asymptotic representation of f(x) as x → + ∞, assuming that we are given certain information about the limiting behaviours of the functions R, N and G.

It is well-known that a homomorphism ø(A→B) between groups A and B induces a homomorphism ø*(ZA→ZB) between the corresponding group rings ZA and ZB over the ring of integers Z. The identical congruence O on B and the unit element eB of B can be characterised by the equations x–y = 0 and x–eB = 0 (x,y ∈ B) respectively. Similarly the congruence Γø corresponding to ø and the corresponding normal subgroup of A are

and {x∈A1 = A,(x–eA)ø = 0} respectively.

Let M be a hypersurface in Euclidean space and let the Ricci curvature of M be bounded below by some nonnegative constant. In this paper, we estimate the sectional curvature of M in terms of the lower bound of Ricci curvature and the upper bound of mean curvature.

In an earlier “Note on Selection from a Normal Multivariate Population” the author considered the transformation of an n-variate normal population induced by “selection” operating to alter the parameters of frequency in a subset of p variates, while preserving normal frequency in the subset. The results were new in notation only; in substance they were first found by K. Pearson.

In an earlier paper, which this note is intended to supplement and in some respects improve, the writer gave a general theorem of duality relating to isobaric determinants with elements Cr and Hr, the elementary and the complete homogeneous symmetric functions of a set of variables. The result was shewn to include as special cases the dual forms of “bi-alternant” symmetric functions given by Jacobi and Naegelsbach, as well as two equivalent forms of isobaric determinant used by MacMahon as a generating function in an important problem of permutations.

The approximation is

We have

Now assume

where A differs from 1/12 by a quantity of the order of 1/n, since the above series for x is convergent; and since n is large, A may be taken to be 1/12.

In (4) Young investigates the representation in terms of his exact seminormal units of substitutional expressions which are unchanged on premultiplication by any permutation of a given set of consecutive letters, or which are changed in sign on premultiplication by an odd permutation of those letters. He illustrates his results by deducing the forms of the matrices representing the positive and negative symmetric groups on a set of consecutive letters.

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined by

the supremum being over all finite sets of disjoint Borel subsets of G.

We examine the action of diffeomorphisms of an oriented surface with boundary on the space of conjugacy classes of SU(2) representations of the fundamental group and prove that in the case of a single periodic diffeomorphism the induced action always has fixed points. For the corresponding 3-dimensional mapping cylinders we obtain families of representations parametrized by their value on the longitude of the torus boundary.