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An example is constructed of a nilpotent group of class 2 and the order of the continuum all of whose abelian subgroups are countable. This answers a question asked by James Wiegold.
We construct two bases of the natural numbers B1 and B2, each of order two, such that (B1 + B2 (n) n¾. Generalisations to sums of bases of order h > 2 are also given.
This paper produces several examples of varieties X for which the global sections functor Γ(X,–): DX‐mod→D(X)‐mod is exact, and makes D(X)‐mod a quotient category of DX‐mod, but is not an equivalence. These varieties are quotients by finite group actions of D‐affine varieties. The torsion of...
We give a short elementary proof of a result of Almkvist and Meurman  on an integrality property of the values taken by the Bernoulli polynomials at a rational number. We use a lemma on the divisibility properties of certain binomial coefficients which seems to be of independent interest.
Let G be a group and let k > 2 be an integer, such that (k2 − 3)(k − 1) < |G|/15 if G is finite. Suppose that the condition |A2| ⩽ k(k + 1)/2 + (k − 3)/2 is satisfied by every it‐element subset A ⊆ G. Then G is abelian. The proof uses the structure of quasi‐invariant sets.
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