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We describe sharply 2-transitive groups whose point stabilizer is a nilpotent {2,3}-group without elements of order 9 and, more generally, in which the third power of each element belongs to the FC-center. In particular, we will prove that these groups are finite. [ABSTRACT FROM AUTHOR]

We consider finite groups G for which any two cyclic subgroups of the same order are conjugate in G. We prove various structure results and, in particular, that any such group has at most one non-abelian composition factor, and this is isomorphic to PSL(2, pm), with m odd if p is odd, or to Sz(22m+1), or to one of the sporadic groups M11, M23, or J1. [ABSTRACT FROM AUTHOR]