Given a self adjoint operator, T, on a Hilbert space H, and given an integer n ≥ 1, we produce a collection , N ∈ L, of n × n positive matrix measures and a unitary map U: such that UTU−1, restricted to the co-ordinate space , is the multiplication operator F(t) → tF(t) in that space. This is a generalization of the spectral representation theory of Dunford and Schwartz, Linear operators, II (1966).

The complemented elements of the lattice of hereditary radicals are characterized. A hypernilpotent complemented hereditary radical is the upper radical determined by a finite number of finite matrix rings. As a corollary, Stewart's characterization of radical semisimple classes is obtained. The methods are universal algebraic in nature.

This paper investigates the normal subgroup structure of the automorphism group Γ of a free abelian group A of countably infinite rank. The finitary automorphisms, that is those acting non-trivially only on a direct summand of A of finite rank, form a normal subgroup Φ of Γ; the sublattice of all normal subgroups of Γ contained in Φ is in fact the sublattice of normal subgroups of Φ and has a quite transparent structure. By contrast there is a profusion of normal subgroups of Γ not contained in Φ. For example, the collection of certain types of these normal subgroups, defined as generalizations of the congruence subgroups of finite dimensional integral linear groups, if partially ordered by inclusion, can be shown to contain infinitely many chains of the order type of the continuum.

A variety of groups is just non-finitely-based if it does not have a finite basis for its laws while all its proper subvarieties do have a finite basis. Recent work of Ol'šanskiiˇ, Vaughan-Lee and Adjan guarantees the existence of at least one just non-finitely-based variety. In this note an infinite number of just non-finitely-based varieties are shown to exist by proving that for every prime p there is a non-finitely based variety of p–groups.

It is shown that the automorphism group of a real Lie algebra operates transitively on the set of its one-dimensional subspaces iff the Lie algebra is abelian, or isomorphic to the algebra of skew-symmetric 3 × 3 real matrices. This allows to conclude that R, S0(2), S0(3) and Spin(3) are the only connected Lie groups such that: (1) the conjugates of every connected set containing e cover a neighbourhood of e, (2) every point sufficiently close to e lies on exactly one 1-parameter subgroup.

Characterizations are obtained of rings R such that the only torsion classes (respectively, hereditary torsion classes) of left unital R-modules are {0} and the class of all modules.

We construct projective planes having non-degenerate homomorphic images, the homomorphisms preserving certain sharply transitive collineation groups.

The Krull Galois theory for infinite separable normal extensions is generalized in this note to non-algebraic extensions. For any extension field E of a field K it is shown that the Galois group G can be given a translation invariant topology such that the closed subgroups are precisely the subgroups that figure in a Galois correspondence. For extension fields E/K such that E/K is of finite transcendence degree and such that E is Galois over each intermediate field the topology turns out to be compact and we have a Galois correspondence in the Krull fashion. For infinite transcendence degree extensions the Galois correspondence remains but compactness is lost. The topology coincides with the Krull topology in the case of algebraic extensions. Further properties of the topology are also studied.

Let p be a prime and the variety of elementary abelian by elementary abelian p-groups. A result of Brisley and Macdonald is generalized as follows. If H is a finite group in and G is a soluble group of p–power exponent such that no section of G is isomorphic to H, then G is nilpotent and its class is bounded by a function of three variables: H, the exponent of G, and the soluble length of G. It is a corollary that if the variety generated by a soluble group G of finite exponent contains , then each finite group in is isomorphic to some section of G.

We give a simplified proof of a general theorem of Chunikhin on existence of subgroups of a finite group. The proof avoids the technical device of “indexials” which Chunikhin set up for this purpose.

A subset S of an additive group G is called a maximal sum-free set in G if (S+S) ∩ S = ø and |S| ≥ |T| for every sum-free set T in G. In this paper, the maximal sum-free sets in cyclic p–groups are characterized to within automorphism.

This paper examines a class of common numerical methods for computing eigenvectors of a compact linear operator. A necessary and sufficient condition is established for every element of an arbitrary given eigenspace to be the limit of a sequence of the approximate eigenvectors obtained by any given method in this class.

One condition on D was omitted from the statement of Theorem B in the paper [1], namely that (and the corresponding condition in Theorem A). The proof of the theorem clearly requires that this be so and it is not difficult to see that such a D can always be chosen. However, Dr R.M. Bryant has pointed out that not only is the theorem true as it stands, but indeed the conditions can be relaxed slightly to allow D to be any group such that

.

Through an editorial error, the second sentence in the penultimate paragraph of [1] does not make sense. The first line in that paragraph should read:

“It now follows that K/G separates the plane, Where K is, as usual,”.