Desmond MacHale

The Predictability of Counterexamples

When teaching mathematics, one is often faced with the task of convincing a student that the converse of a given theorem is false or that the theorem is as strong as possible. The usual method involves the production of a counterexample. However, even a quick glance at a wide range of textbooks shows a singular lack of variety in the counterexamples exhibited, arising no doubt from a lack of imagination on the part of the authors. Indeed, an inexperienced student might be led to conjecture that the following statements are theorems: 1. The function f(x) = IxI is the only real function that is continuous but not differentiable. 2. The real interval [0,1] is the only uncountable set. 3. The function defined on [0,1] by

Groups with an automorphism cubing many elements

Let G be a group and α n the mapping which takes every element of G to its n th power, where n is an integer. It is well known that if α n is an automorphism then G is Abelian in the cases n = -1,2, and 3. For any other integer n (≠ 0) there exists a non-Abelian group which admits α n as the identity automorphism. Indeed Miller (1929) has shown that if n ≠ 0, ±1, 2, 3 then there exist non-Abelian groups which admit α n as a non-trivial automorphism.

Odd order groups with an automorphism cubing many elements

We determine the structure of a nonabelian group G of odd order such that some automorphism of G sends exactly (1/p)|G| elements to their cubes, where p is the smallest prime dividing |G|. These groups are close to being abelian in the sense that they either have nilpotency class 2 or have an abelian subgroup of index p .

Variations on a Theme: Rings Satisfying x 3 = x Are Commutative

Abstract A ring satisfying x3 = x is necessarily commutative. We consider a variety of weaker forms of this condition and show that many, but not all of them, imply commutativity. We also present a variety of elementary proofs of the fact that x3 = x implies commutativity.

That 3-4-5 Triangle Again

A number of authors have considered the following (and related) problems: Find all integer-sided triangles such that the perimeter equals the area. In the March 1981 edition of Mathematics Teacher Lee Markowitz gives the following full set of solutions (though not the considerable calculations involved): (5, 12, 13) (6, 8, 10) (6, 25, 29) (7, 15, 20) (9, 10, 17). However, in many formulae involving the triangle, it is the semi-perimeter rather than the perimeter that arises naturally.

Powers of Commutators and Anticommutators

For , we give elementary proofs of commutativity of rings in which the identity holds for all commutators . For even , we show that the commutativity of rings satisfying such an identity is equivalent to the anticommutativity of rings satisfying the corresponding anticommutator equation.

Dividing an Angle into Equal Parts

The problem of finding a geometric construction in a finite number of steps to trisect an arbitrary angle using straightedge and compasses alone is a very old one, originally proposed by the mathematicians of ancient Greece. It was only with modern algebraic techniques in the nineteenth century that it was shown conclusively that no such construction can exist. This fact has in no way deterred a legion of crank angle-trisectors from presenting their alleged solutions to the problem! However some angles (even non-constructible ones) can be trisected: we’ll see later, for example, that given an angle π/7 it can be trisected. We can also consider the more general problem: