Let n be a positive integer and let G be a group. We denote by , (G) a certain extension of the non-abelian tensor square G G by G ,G. Set T if the size of the conjugacy class , , , (G) = f g h j g,h 2 G g. We prove that , (G) x , , ,,n for every x 2 T (G), then the second derived subgroup , (G)′, ′,is , nite with n-bounded order. Moreover, we obtain a suffi, cient condition for a group to be a BFC-group.

M24 is the largest Mathieu sporadic simple group of order 244823040=210.33.5.7.11.23 contains all the other Mathieu sporadic simple groups as subgroups. The object in this paper is to study the ranks of M24 with respect to the conjugacy classes of all its nonidentity elements.

We survey known results concerning how the conjugacy classes contained in a normal subgroup and their sizes exert an inuence on the normal structure of a finite group. The approach is mainly presented in the framework of graphs associated to the conjugacy classes, which have been introduced and developed in the past few years. We will see how the properties of these graphs, along with some extensions of the classic Landau's Theorem on conjugacy classes for normal subgroups, have been used in order to classify groups and normal subgroups satisfying certain conjugacy class numerical conditions.

Please click on PDF to view the abstract.