We consider ineducible Goppa codes of length qm over Fq defined by polynomials of degree r, where q = pt and p, m, r are distinct primes. The number of such codes, inequivalent under coordinate permutations and field automorphisms, is determined.

Given polynomials a and b over an integral domain R, their tensor product (denoted a ⊗ b) is a polynomial over R of degree deg(a) deg(b) whose roots comprise all products αβ, where α is a root of a, and β is a root of b. This paper considers basic properties of ⊗ including how to factor a ⊗ b into irreducibles factors, and the direct sum decomposition of the ⊗-product of fields.

Let G be a finite group that acts on a finite group V, and let p be a prime that does not divide the order of V. Then the p-parts of the orbit sizes are the same in the actions of G on the sets of conjugacy classes and irreducible characters of V. This result is derived as a consequence of some general theory relating orbits and chains of p-subgroups of a group.

The main result is that every torsion-free locally nilpotent group that is isomorphic to each of its nonnilpotent subgroups is nilpotent, that is, a torsion-free locally nilpotent group G that is not nilpotent has a non-nilpotent subgroup H that is not isomorphic to G.

We provide an upper bound for the order of a nilpotent injector of a finite solvable group with Fitting subgroup of order n. We also show that the same bound is an upper bound for the number of conjugacy classes, provided that the k(G V)-conjecture holds for solvable G all primes dividing n.

In this paper we investigate subtractive varieties of algebras that are Fregean in order to get structure theorems about them. For instance it turns out that a subtractive variety is Fregean and has equationally definable principal congruences if and only if it is termwise equivalent to a variety of Hilbert algebras with compatible operations. Several examples are provided to illustrate the theory.

A pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.

For a maximal coaction δ of a discrete group G on a C*-algebra A and a normal subgroup N of G, there are at least three natural A × G ×δ| N - A ×δ|1 G/N imprimitivity bimodules: Mansfield's bimodule ; the bimodule assembled by Ng from Green's imprimitivity bimodule and Katayama duality; and the bimodule assembled from and the crossed-product Mansfield bimodule . We show that all three of these are isomorphic, so that the corresponding inducing maps on representations are identical. This can be interpreted as saying that Mansfield and Green induction are inverses of one another ‘modulo Katayama duality’. These results pass to twisted coactions; dual results starting with an action are also given.

In this paper we investigate subtractive varieties of algebras that are congruence quasi-orderable. Though this concept has its origin in abstract algebraic logic, it seems to be worth investigating in a purely algebraic fashion. Besides clarifying the algebraic meaning of this notion, we obtain several structure theorems about such varieties. Also several examples are provided to illustrate the theory.