Graham Higman published two important papers in 1960. In the first of these papers he proved that for any positive integer n the number of groups of order pn is bounded by a polynomial in p, and he formulated his famous PORC conjecture about the form of the function f(pn) giving the number of groups of order pn. In the second of these two papers he proved that the function giving the number of p-class two groups of order pn is PORC. He established this result as a corollary to a very general result about vector spaces acted on by the general linear group. This THEOREM takes over a page to state, and is so general that it is hard to see what is going on. Higman's proof of this general THEOREM contains several new ideas and is quite hard to follow. However in the last few years several authors have developed and implemented algorithms for computing Higman's PORC formulae in special cases of his general THEOREM. These algorithms give perspective on what are the key points in Higman's proof, and also simplify parts of the proof.

We give a new proof of Glauberman's ZJ THEOREM, in a form that clari, es the choices involved and off, ers more choices than classical treatments. In particular, we introduce two new ZJ-type subgroups of a p-group S, that contain ZJr(S) and ZJo(S) respectively and can be strictly larger.

Let p be a prime number and let n be a positive integer prime to p. By an Ihara-result, one means the existence of an injection with torsion-free cokernel, from a full lattice, in the space of p-old modular forms, into a full lattice, in the space of all modular forms of level np. In this paper, Ihara-results are proven for genus two Siegel modular forms, Siegel-Jacobi forms and Hilbert modular forms. Ihara did the genus one case of elliptic modular forms[1]. A geometric formulation is proposed for the notion of p-old Siegel modular forms of genus two, using clarifying comments by R. Schmidt[2] and, then, following suggestions in an earlier paper[3] on how to prove Ihara results. The main THEOREM in[3] is used, where an argument by G. Pappas has been extended to prove the torsion-freeness of certain cokernel, using the density of Hecke-orbits in the moduli space of principally polarized abelian varieties and in the Hilbert-Blumenthal moduli space, which was proved by C. Chai[4].

Let G be an abelian group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper, we generalize the graded primary avoidance THEOREM for modules to the graded primal avoidance THEOREM for modules. We also introduce the concept of graded PL-compactly packed modules and give a number of its properties.

Introduction: In the last few years there is a growing interest in the theory of quasi-metric spaces and other related structures such as quasi-normed cones and asymmetric normed linear spaces, because such a theory provides an important tool in the study of several problems in theoretical computer science, approximation theory, applied physics, convex analysis and optimization. Many works on general topology and functional analysis have recently been obtained in order to extend the well-known results of the classical theory of normed linear spaces to the framework of asymmetric normed linear spaces and quasi-normed cones. An abstract cone is analogous to a real vector space, except that we take R+ as the set of scalars. In 2004, O. Valero introduced the normed cones and proved some closed graph and open mapping results for normed cones. Also Valero defined and studied some properties of quotient normed cones. P. Selinger studied the norm properties of a cone with its order properties and proved Hahn-Banach THEOREMs in these cones under the appropriate conditions. Valero and his colleagues discussed the metrizability of the unit ball of the dual of a normed cone and the isometries of normed cones. Other properties are investigated in a series of papers by Romaguera, Sanchez Perez and Valero. The Bishop-Phelps THEOREM is a fundamental THEOREM in functional analysis which has many applications in the geometry of Banach spaces and optimization theory. The classical Bishop-Phelps THEOREM states that “ the set of support functionals for a closed bounded convex subset B of a real Banach space X, is norm dense in X* and the set of support points of B is dense in the boundary of B". Indeed, E. Bishop and R. R. Phelps answer a question posed by Victor Klee in 1958. We give an analogue to the normed cones, in fact we show that in a continuous normed cone the set of support points of a closed convex set is a dense subset of the boundary under the appropriate hypothesis. Conclusion: In this paper the notion of support points of convex sets in normed cones is introduced and it is shown that in a continuous normed cone, under the appropriate conditions, the set of support points of a bounded Scott-closed convex set is nonempty. We also present a Bishop-Phelps type THEOREM for normed cones.

We give a new proof of the well known Wedderburn's little THEOREM (1905) that a finite division ring is commutative. We apply the concept of Frobenius kernel in Frobenius representation THEOREM in finite group theory to build a proof.

Let T be a compact Hausdorff topological space and let M denote an n–dimensional subspace of the space C(T ), the space of real–valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii [7] attributes the Basic THEOREM to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel’man, which uses Helly’s THEOREM, now the paper obtains a new proof of the Basic THEOREM. The significance of the Basic THEOREM for us is that it reduces the characterization of a best approximation to f Î C(T ) from M to the case of finite T , that is to the case of approximation in l¥(r). If one solves the problem for the finite case of T then one can deduce the solution to the general case. An immediate consequence of the Basic THEOREM is that for a finite dimensional subspace M of C0(T ) there exists a separating measure for M and f Î C0(T)\M, the cardinality of whose support is not greater than dimM+1. This result is a special case of a more general abstract result due to Singer [5]. Then the Basic THEOREM is used to obtain a general characterization THEOREM of a best approximation from M to f Î C(T ). We also use the Basic THEOREM to establish the sufficiency of Haar’s condition for a subspace M of C(T ) to be Chebyshev.

The generalized principal Ideal THEOREM is one of the cornerstones of the dimension theory for the Noetherian rings. For an R-module M, certain submodules of M that play a role analo- gous to that of prime ideals in the ring R are identied. Using this denition, we extend the this THEOREM to modules.