On finite groups in which cyclic subgroups of the same order are conjugate

Similar kinds of problems have often been object of investigation. For instance, in [5] Fitzpatrick, using the classification of finite simple groups, proved that if in a finite group G any two elements of the same order are conjugate, thenG is isomorphic with the symmetric group Sn, with n ∈ {1, 2, 3} (see also [4]). Then in [10] there is the classification of finite groups for which elements of the same order are conjugate or inverse-conjugate. Similar results, but concerning fusion in Aut G, have been obtained in [20], [11] and [12]. In [17], the author considers finite groups G for which any two isomorphic subgroups are conjugate in G. The main result of the present paper is the following Theorem. Let G be a finite csc-group. Then