An infinite group with the weak wide commensurable property is shown to be abelian, provided that it is locally finite or locally graded or non-perfect or linear. We also investigate the properties of infinite non-abelian groups with the weak wide commensurable property. | Turk J Math 29 (2005) , 403 – 412. ¨ ITAK ˙ c TUB On Groups with the Weak Wide Commensurable Property ¨ Ay¸se Berkman, Mahmut Kuzucuo˘glu, Erdal Ozyurt Dedicated to Prof. Dr. Cemal Ko¸c on his 61st birthday Abstract An infinite group with the weak wide commensurable property is shown to be abelian, provided that it is locally finite or locally graded or non-perfect or linear. We also investigate the properties of infinite non-abelian groups with the weak wide commensurable property. Moreover, we describe completely the structure of infinite locally finite groups whose p-subgroups have the weak wide commensurable property. (AMS MSC: 20F50, 20E34). 1. Introduction If a group-theoretical property of groups is common to all finite groups, then it is called a finiteness property. Some well-known examples of finiteness properties are: being finitely generated, locally finite, residually finite, FC, min and max conditions. For details, the reader might like to see [7, Chapter 14]. We study infinite groups that satisfy a particular finiteness property, namely the weak wide commensurable property. To be precise, we consider infinite groups in which any two non-trivial proper subgroups have commensurable conjugates. Recall that two subgroups are called commensurable if their intersection is of finite index in both subgroups. Clearly all finite groups, quasi-cyclic p-groups for every prime p and the additive group of integers satisfy the weak wide commensurable property. A more interesting class is 403 ˘ ¨ BERKMAN, KUZUCUOGLU, OZYURT quasi-finite groups (these are groups whose proper subgroups are all finite). Probably the most well-known non-abelian quasi-finite group is the Tarski group which was constructed by Ol’shanskii, answering the question of Tarski on the existence of infinite groups whose non-trivial proper subgroups are of order a fixed prime. For details, see [6, Theorem ]. There is also a non-abelian torsion-free group that satisfies the weak wide .
