[GAP Forum] Does every finite group have a built-in total ordering?

[Will Chen]

Dear GAP Forum,

I've been writing code that given a finite group G, computes actions on the
set of conjugacy classes of generating pairs of G. To implement this action
I've been working with "minimum representatives" of each conjugacy class -
ie, given a generating pair (g,h) of G, I instead will work with the pair
(g',h'), defined as follows:

g' := Minimum(List(ConjugacyClass(G,g)));
t := RepresentativeAction(G,g,g');
C := Centralizer(G,g);
h' := Minimum(List(Orbit(C,h^t)));

Thus, (g',h') is the "minimum" element of the conjugacy class of (g,h),
under the lexicographic ordering built on the internal ordering on G which
is used by the function "Minimum".

I've built a bunch of sanity checks into my code, and so far there hasn't
been any issues, but I wonder - is it possible that I've just been lucky?
(I've tried using "safer" implementations of the action, but they generally
run slower than just operating on minimum objects as defined above - unless
anyone has any ideas?)

In other words, given a finite group G represented in GAP, is G guaranteed
to have an immutable total ordering which "Minimum" is always guaranteed to
use when called via "Minimum(List(X))" where X is a subset of G?

- Will

-- 

William Chen
Member, School of Mathematics
Institute for Advanced Study,
Princeton, NJ, 08540
oxeimon at gmail.com