If one were to lay out the dihedral group $D_4$ multiplication table, it might
look like this (generators $a$ and $b$):
If we abstract from the generators and just call the elements $0,1,2,3,4,5,6,7$,
we obtain a raw multiplication table for $D_4$ (which I do not display). My question is:
Q. Is there a canonical, preferred multiplication table using just integers
to indicate the group elements for a finite group, or is the ordering of elements
essentially arbitrary?
I am thinking of comparing multiplication tables of groups of the same finite order,
and I am wondering if I can compare canonical tables, or should I instead compare
all possible permutations of the tables?
Best Answer
One can usefully impose a "canonical" ordering on these tables by selecting (for instance) the lexicographically least table in its isomorphism class. This has been done in some programs for generating all such tables up to isomorphism satisfying some given set of (typically equational) constraints. If you already have all the tables in hand then, depending upon what you are going to do with them, it may or may not be worth the cost of computing it.
Of course, this choice isn't really canonical at all; it is just a useful convention, and a different one might suit your purposes just as well, or better.
From a group-theoretic perspective, having "anonymised" the group elements by replacing them with integers, one ordering is pretty much as good as any other.