Footnote humor

I found today an enjoyable little footnote in Elliptic Tales, a recent book on elliptic curves and number theory written by Boston College professors Avner Ash and Robert Gross. The book is a tough and ambitious undertaking with the goal of explaining one of the most difficult areas of mathematics to a general readership. Even though I have a little more mathematical training than a lay reader, the book has been slow going, partly because of the difficult subject and partly because real life events have been too important to spare any reading time at all.

In the middle of the book, the authors define abelian groups. Briefly, an abelian group is a set of elements G endowed with a mathematical operation R that satisfies five properties:

  1. Closure: If a, b lie in G, then a R b is also in G.
  2. Associativity: (a R b) R c = a R (b R c).
  3. Symmetry: a R b = b R a.
  4. Existence of a neutral element z such that a R z = a.
  5. Existence of an “inverse” element, such that for every a, there exists -a, such that a R (-a) = z.

From there, the authors go on to describe a generator subgroup. This is a subset of G and consists of a few elements from G that can be combined using the operation R to generate the all elements of G. For example, if R is taken to be the addition operation, then the elements of the set {2,5} can be combined according to 2m + 5n to generate the entire set of integers using appropriate values for m and n. Here, m and n are not really multipliers (since the operation of multiplication hasn’t been explicitly defined) but just a shorthand way of writing 2 + 2 + … m times + 5 + 5 + … n times. Thus {2, 5} is a generator because 2 and 5 are relatively prime. With this explanation on generators came a sly footnote that made me smile:

We make the convention that the one-element group is generated by nothing at all. Since 0 has to be there, it doesn’t need to be generated. Theological analogies will be left to the reader.

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