William, David, and Jack,
Many thanks for all the excellent suggestions. Initially, I'll use
Jack's local variable trick, though as David says a more complete
implementation would probably want to keep track of the parent free
group. There's also a clean multistep way that doesn't introduce any
variables in GAP:
sage: F = gap.new("FreeGroup(2)")
sage: G = F.FactorGroupFpGroupByRels([F.1*F.2*F.1**-1*F.2**-1])
Properly wrapping finitely presented groups would be a slightly tricky
task, and, unfortunately, this is not my current goal. There's a lot
more to a fp-group besides just the generators and relations, e.g.
information about when a group is a subgroup of another group which is
need for comparing subgroups, computing homorphisms between groups,
etc. One could, of course, just delegate most of these issue to GAP,
and that's certainly a reasonable approach to add finitely presented
groups to Sage. Unfortunately, for what I typically do (e.g. find
low-index subgroups), Magma is *much* faster than GAP, so I'm only
aiming for a finitely presented group class which allows me to create/
keep track of fp-groups at the Python level, and will just have to
drop into the Magma interpreter for most of actual computational
work.
Best,
Nathan
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