A simple group G cannot contain a central involution t with CG(t) = 〈t〉 × M, where M is isomorphic to a simple Mathieu group.

We introduce some generalizations of Kadec'-Klee norms and use them to study characteristics of subspaces of conjugate spaces and smoothness. We give some connections between such characteristics and basic sequences, which yield, in particular, sharpenings and simpler proofs of some known characterizations of reflexivity.

Let λ(G) be the cardinality of a maximal sum-free set in a group G. Diananda and Yap conjectured that if G is abelian and if every prime divisor of |G| is congruent to 1 modulo 3, then λ(G) = |G|(n−1)/3n where n is the exponent of G. This conjecture has been proved to be true for elementary abelian p−groups by Rhemtulla and Street ana for groups by Yap. We now prove this conjecture for groups G = Zpq ⊕ Zp where p and q are distinct primes.

This paper finds restrictions on the parameters of supplementary difference sets in any group G with a subgroup of index 2, which therefore includes all cyclic groups of even orders. As a corollary to the main theorem, we have that if S1, …, Sr are r − {2v, k1, …, kr; 2λ} supplementary difference sets in such a group, then not all of v, k1, …, kr, λ are odd; also is the sum of r squares.

It has been incorrectly asserted that each non-trivial semi-simple radical class of associative rings is a variety defined by an equation of the form xn = x. In this paper we give, for each non-trivial semi-simple radical class of associative rings, a set of equations which does define that class as a variety.

László Babai raised the question whether every infinite group G contains a set B of elements, of cardinal equal to the order of G, such that for elements a, b, c of B the equation a = bc−1b implies a = c. We show that the answer is affirmative for certain classes of groups, including the soluble groups and the countable groups, if the Axiom of Choice is assumed, but negative, even for abelian groups, if a different axiom, compatible with Zermelo-Fraenkel set theory but incompatible with the Axiom of Choice, is assumed.

Christine W. Ayoub has defined diagrams for abelian groups and has shown that the structure of an abelian group A is determined by certain subgroups Ti of A which arise naturally from a given diagram for A. In this paper it is shown that a given diagram for A is formed from certain elementary diagrams for the Ti.

If (A′, B′), (B′, C′) and (A, B), (B, C) are torsion-torsion free theories on RM and MR respectively which are generated by an idempotent ideal I of R, then M ∈ RM is said to be relatively flat if (·) ⊗ RM preserves short exact sequences 0 → L → X → N → 0 in MR with N ∈ B. Several characterizations of relatively flat modules are given and it is shown that any module M ∈ RM which is codivisible with respect to (A′, B′) is relatively flat. In addition, when (A′, B′) is hereditary, it is proven that M ∈ RM is relatively flat if and only if M/IM is a flat R/I-module. Finally, a relatively flat dimension for M ∈ RM and a left global relatively flat dimension for R are defined and it is shown, again when (A′, B′) is hereditary, that the left global relatively flat dimension of R coincides with the left global flat dimension of R/I.

Let a(z) and b(w) be two rational functions in z or W with algebraic coefficients, where a(0) = 0 and let

Assume that 0 < |z| < 1, that z is not a pole of for n ≥ 0, that w is neither a pole nor a zero of b(w, n) for n ≥ 1, and that the series

for fixed w is a transcendental function of z. Then, if z and W are algebraic numbers, f(z, w) is a transcendental number.

The sufficient optimality conditions, of Fritz John type, given by Gulati for finite-dimensional nonlinear programming problems involving polyhedral cones, are extended to problems with arbitrary cones and spaces of arbitrary dimension, whether real or complex. Convexity restrictions on the function giving the equality constraint can be avoided by applying a modified notion of convexity to the other functions in the problem. This approach regards the problem as optimizing on a differentiable manifold, and transforms the problem to a locally equivalent one where the optimization is on a linear subspace.

In this note a converse duality theorem is proved for a class of nonlinear programming problems over polyhedral cones in finite dimensional complex space by a direct use of a Kuhn-Tucker type necessary and sufficient condition for constrained optimization in complex space.

A system of set theory which appears as an extension of Ackermann set theory is introduced. In this sytem we construct a syntactic model for a theory proposed by Feferman for the development of category theory.

Let Γ be a family of isoclinic finite groups, and let acl(Γ) be the group of autoclinisms of Γ. In this paper we prove the following formula:

where the sum is taken over a complete system of stem groups S in Γ. This result is due to P. Hall, who outlined a proof in his paper, “On groups of automorphisms” (J. reine angew. Math). 182 (1940), 194–204, using presentations, whereas in this paper we consider stem groups in terms of central extensions.

Recently, Eklund (Proc. Amer. Math. Soc. 47 (1975), 137–142) has shown that to each continuous function F on ∂pQ −⊲ {∂Ω × [0, T]} ∪ {Ω × (0)} there is a unique solution to the boundary value problem

where L is a linear second order parabolic operator in divergence form, Ω ⊂ Rn is a bounded domain with compact closure and ∂Ω denotes its boundary. In this note, it is shown that the existence theorem of Eklund remains valid for the following boundary problem

The purpose of this paper is to completely characterize splitting torsion theories over commutative rings. In particular, if (T, F) is a torsion theory for which T(R) = 0, then (T, F) is a splitting theory if and only if T contains only a finite number of nonisomorphic simple modules and every module in T is semisimple injective. In addition, an ideal theoretic characterization of splitting torsion theories is given, of which one consequence is that splitting torsion theories are TTF; furthermore, if R is also noetherian, then such torsion theories are centrally splitting. The known theorems concerning the splitting of the Goldie and simple torsion theories (for commutative rings) are derived from our theorem.