tailieunhanh - On groups with the weak wide commensurable property
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 .
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