Group names

Recently, I discussed Olexandr Konovalov’s crowd-sourced project to verify and extend the known values of the function gnu(n), the number of groups of order n.

A month and a half ago (but I have only just noticed it), Olexandr raised a related question; I would like to say something about it.

We can’t give every group a name, although there have been valiant attempts to do so. There are nearly 50 billion groups of order 1024; human-readable names for all of them are unlikely to be much shorter or more informative than, say, power-commutator presentations for these groups.

However, there are certain common groups which already have names. Among them are the dihedral groups, and here the problem arises. Some people use D_n to denote the dihedral group of order 2n (the group of symmetries of the regular n-gon); while some use this notation (for even n) for the dihedral group of order n.

Both approaches are widely used, and both are instances of common conventions for naming groups. For the symmetric groups, the parameter indicates the “size” of the object the group naturally acts on: we say S₈, not S₄₀₃₂₀. On the other hand, Dickson refers to the simple group of order 25920 (the simple normal subgroup of the group of the 27 lines in a cubic surface) as G₂₅₉₂₀, and subsequent authors such as J. A. Todd and J. S. Frame followed to some extent, though now we usually call it either PSp(4,3) or PSU(4,2).

Olexandr points to different usages in different computer algebra systems, which is certainly a confusion and a source of possible error for users. He advocates OpenMath as the way out of the difficulty.

But on the subject, you may have spotted another example of the same thing just above. The group PSU(4,2) is the projective (this means, central quotient of) special (matrices of determinant 1) unitary group of 4×4 matrices over the field of 4 elements. So why isn’t it PSU(4,4)? Well, to many people, it is. (They probably write it as PSU(4,2²) to try to disambiguate it – unitary groups are only defined over fields with an automorphism of order 2, and for finite fields this means square order – but this is really only a kludge.)