# FOM: computable ordered nonabelian group?

JoeShipman@aol.com JoeShipman at aol.com
Tue Nov 6 18:02:48 EST 2001

Matt,

You ask for an example of a computable ordered nonabelian group.  I did a quick search and came across the following abstract.  Either this is the example you are looking for, or else it is a group with an unsolvable word problem in which case it would be by far the simplest known such group!  Since the abstract refers to "direct calculation" I suspect the full paper will reveal that the group has a solvable word problem and is your desired example.

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International Conference on Ordered Algebraic Structures and Related Areas
July 6 - 10, 1998
Nanjing University
Nanjing, People's Republic of China
Conference Organizers
W.C.Holland, P.Conrad, K.Denecke, M.Giraudet, A.C.Kim, V.Kopytov, N.Y.Medvedev, I.Rival, K.P.Shum, Dao-Rong Ton, C.Tsinakis, Minxie Jiang and Kaiyao He

Ordered Groups in which Every Automorphism Preserves the Order
presented by
Vasiliy Bludov
Irkutsk State University, Professor, Head of the Chair of Algebra, Logic and Cybernetics
In the report we represent an example of orderable nilpotent group G with the property: every automorphism of G preserve every full order of G . We use the following notations: AutG --- group of all automorphisms of G , IAG --- subgroup of IA-automorphisms (i.e. subgroup of automorphisms induce identical automorphism on G/G' ).
Example. Let k,q --- nonzero integers and Gk,q --- nilpotent group of class 5 with generators a , b and defining relations:
[a,[a,[a,[a,b]]]]=[a,[a,b,b,b]]=[a,b,b,b,b]=1,
[a,[a,[a,b,b]]]=[a,[a,b],[a,b]],
[a,[a,[a,b]]]=[a,[a,b],[a,b]]k=1,
[a,b,b,b]=[a,b,[a,b,b]]q.
Direct calculations show that Gk,q is orderable, AutGk,q=IAGk,q and every automorphism of Gk,q preserve every full order of Gk,q . Moreover, \mbox{IA }Gk,q is orderable as torsion-free nilpotent group. Remark. If G is a two-generated torsion-free group of nilpotency class less than 5 then AutG has nontrivial elements of finite orders.
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-- Joe Shipman