# FOM: computable ordered nonabelian group?

Matthew Frank mfrank at math.uchicago.edu
Tue Nov 6 14:11:00 EST 2001

```Does anyone know a computable example of an ordered nonabelian group?
(i.e.:  The group composition law and the order should both be computable,
and the order should be total and left-and right- invariant.)

Or is there a proof that there is no such example?

--Matt

P.S.  Here's a standard example which is not obviously computable:
(from A.M.W. Glass, Partially Ordered Groups, 1999)

Take G = the free group on two generators.
Define a descending series of subgroups G_i by
G_0 = G
G_(i+1) = subgroup generated by all commutators xy(x^-1)(y^-1),
with x in G_i and y in G.

Each G_i/G_(i+1) is abelian and torsion-free, so orderable.

For every g not equal to 1, there is a unique m such that
g is in G_m but not G_(m+1); for such m, define
g>1 in G iff g>1 in G_m/G_(m+1)

But I see no obvious way to compute m, since I see no obvious way to check
membership in the G_i, and I suspect that it's impossible by the
unsolvability of the word problem.

```