Abelian groups

This post is inspired by a very nice paper by Henrik Kragh Sørensen in the current issue of the Bulletin of the British Society for the History of Mathematics, entitled “What’s Abelian about abelian groups?” Abel worked and died long before the modern definition of groups was given, so why do we name commutative groups after him?

It turns out that the name abelsche Gruppen was adopted by Weber in his algebra book of 1895/1896, but was already in use by this time: Kronecker had used the term “abelsche” for a class of equations, and it was transferred to the Galois groups of these equations when the connection was understood in the second half of the nineteenth century.

Based on his early work on the unsolvability by radicals of the quintic and his later work on dividing an arc of a curve such as the lemniscate into n equal parts, Abel had shown the following. If an irreducible equation of degree n has the two properties

  • all its roots xi can be expressed as rational functions φi(x0) of a given root x0;
  • the functions φi commute under composition; that is, φij(x)) = φji(x)) for all i,j;

then the equation can be solved by radicals.

In modern terminology, the assumptions say that the Galois group of the polynomial acts regularly on the roots (that is, any one of them generates a splitting field of the polynomial) and is an abelian group. The first assumption loses no generality; any finite normal separable extension can be generated by a single element.

So it seems clear that the attribution to Abel is justified. Interestingly, equations with cyclic groups were called “simply abelian” for a time (Abel investigated these also).

This leaves unanswered a mystery which has puzzled me for many years. In his book Linear Groups: With an Exposition of the Galois Field Theory published in 1901, Dickson uses the term “Abelian group” for what we now call “symplectic group”, a group preserving a nondegenerate alternating bilinear form on a vector space. Far from being “abelian”, these groups are close to being simple! I have not tried to think it through carefully, but I imagine that, at least for prime values of n, they arise as Galois groups in the problem of dividing the periods of suitable elliptic functions by n. Dickson went on to define “hypoabelian groups”, which are orthogonal groups in characteristic 2 (where they are subgroups of the corresponding symplectic groups, because a quadratic form in characteristic 2 polarises to an alternating bilinear form which is preserved by the corresponding orthogonal group).

Dickson’s book is a classic, and is still the source of choice for the list of subgroups of the groups PSL(2,q), or LF(2,q) as Dickson called them (for “linear fractional”).

So I’d like to know the answers to a couple of questions:

  • Did Dickson know about Weber’s book or the earlier work of Kronecker?
  • Why, precisely, did he use the term “abelian groups” for the symplectic groups?
  • When did the present usage become standard?

There are other interesting questions about group-theoretic terminology. What we now call “p-groups” were, to Burnside, “groups whose orders are powers of a prime”, and “nilpotent groups” were, if memory serves me well, “groups with properties analogous to those of groups whose orders are powers of a prime”. It was clear that a new term was needed, but who introduced it, and when?

(Incidentally, the term “p-groups” is unfortunate, for various reasons:

  • the Klein four-group is a 2-group;
  • in design theory, there are t-designs and λ-designs; so is a 2-design a t-design with t = 2, or a λ-design with λ = 2? These are quite different things.)

Again, why did the different terms “soluble” and “solvable” come to be used for the same class of groups in British and American English?

I mentioned the invention of the term “loop” for a certain generalisation of a group by Albert, inspired by the Chicago Loop, here. It seems that Albert was dissuaded from calling them “knots”, probably fortunately.


About Peter Cameron

I count all the things that need to be counted.
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4 Responses to Abelian groups

  1. David Roberts says:

    There are also n-groups, where n has nothing to do with p; n can take integer values in the range [1,\infty]. I have a paper called “The inner automorphism 3-group of a strict 2-group”, which would make little sense to a person used to p-groups!

    • Toby Bartels says:

      One may distinguish p-groups and n-groups using the longer terms ‘p-primary group’ and ‘n-dimensional higher group’; I managed to get those prominently placed on the relevant Wikipedia articles, so from there they may catch on.

  2. One of the problems with p-groups is that the term is used both generically (p-group = group of unspecified prime power order, or sometimes group of p-power order where p is specified earlier) and specifically; the latter usage is more problematic. What is a 2-group?

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