B. H. Neumann

On Finite Groups With ‘Hidden’ Primes

The starting point of this investigation was a question put to us by Martin B. Powell: If the prime number p divides the order of the finite group G , must there be a minimal set of generators of G that contains an element whose order is divisible by p ? A set of generators of G is minimal if no set with fewer elements generates G . A minimal set of generators is clearly irredundant, in the sense that no proper subset of it generates G ; an irredundant set of generators, however, need not be minimal, as is easily seen from the example of a cyclic group of composite (or infinite) order. Powells question can be asked for irredundant instead of minimal sets of generators; it turns out that the answer is not the same in these two cases. A different formulation, together with some notation, may make the situation clearer.

Ensuring commutativity of finite groups

Comments are made on the following question. Let m, n be positive integers and g a finite group. Suppose that for all choices of a subset of cardinality m and of a subset of cardinality n in g some member of the first commutes with some member of the second. Under what conditions on m, n is the group abelian?

Monotone functions mapping the set of rational numbers on itself

The functions f defined by or by for c rational and less than + 1 map the set of rational numbers between 0 and 1 one-to-one onto itself; and they are the only fractional linear functions with this property. Miss Tekla Taylor recently raised the question * whether these are the only differentiable functions with the stated property. In the present note we show, by two different constructions, that the answer is negative; in each case much freedom remains, which could be used to make the functions in question have various additional properties.

On Characteristic Morphisms: In Memoriam Thomas Macfarland Cherry

This note is concerned with a translation of some concepts and results about characteristic subgroups of a group into the language of categories. As an example, consider strictly characteristic and hypercharacteristic subgroups of a group: the subgroup H of the group G is called strictly characteristic in G if it admits all ependomorphisms of G ; that is all homomorphic mappings of G onto G ; and H is called hypercharacteristic 2 in G if it is the least normal subgroup with factor group isomorphic to G/H , that is if H is contained in every normal subgroup K of G with G/K ≅ G/H .