The study of orderable groups is a topic that is all too often overlooked as a topic in algebra. The subject of orderable groups is a field of study which is directly associated with algebraic group theory, algebraic topology, and set theory. This paper will act as a guide into the world of orderable groups. It begins by enlightening the reader to the fundamental axioms of orderable groups, as well as, noting various important groups on which orders are often considered. We will then consider more interesting groups, on which the placement of orders is considered less often. After considering the orderings placed on various groups, we wish to consider in further detail the topologies of the sets of these orders. In particular, it is important to consider whether the set of orders placed on a particular group is finite or uncountable. We prove the latter by creating a homeomorphism from the group to the Cantor set, a set which is known for its uncountability. / Master of Science

Nous pre´sentons une condition sur les groupes de Garside que nous appelons la structure de Dehornoy. Une ite´ration d’une telle structure conduit a` une ordre a` gauche sur le groupe. Nous montrons des conditions pour qu’un groupe de Garside admet une structure de Dehornoy, et nous appliquons ce crite`re pour prouver que les groupes d’Artin de type A et I2(m), m ≥ 4, ont des structures de Dehornoy. Nous montrons que les ordres a` gauche sur les groupes d’Artin de type A obtenus a` partir de leurs structures de Dehornoy sont les ordres de Dehornoy. Dans le cas des groupes d’Artin du type I2(m), m ≥ 4, nous montrons que les ordres a` gauche de´rive´es de leurs structures de Dehornoy co¨ıncident avec les ordres obtenus a` partir des plongements de ces groupes dans les groupes de tresses. / We introduce a condition on Garside groups that we call Dehornoy structure. An iteration of such a structure leads to a left order on the group. We show conditions for a Garside group to admit a Dehornoy structure, and we apply these criteria to prove that the Artin groups of type A and I2(m), m ≥ 4, have Dehornoy structures. We show that the left orders on the Artin groups of type A obtained from their Dehornoy structures are the Dehornoy orders. In the case of the Artin groups of type I2(m), m ≥ 4, we show that the left orders derived from their Dehornoy structures coincide with the orders obtained from embeddings of the groups into braid groups.