improving the results of i. m. isaacs on derived subgroups and centers of capable groups

some results on derived subgroups and centers of capable groups are given by i. m. isaacs. the goal of this talk is to improve these results under the weaker hypothesises. moreover, we show that there exists an upper bound for the index of the center of 2-generated finite p-groups with cyclic derived subgroup in terms of the order of its derived subgroup. a group g is called capable if there exists a group h such that g ≌ h/z(h). capability plays an important role in p. hall’s classification scheme for p-groups up to isoclinism. it is really very interesting to find the relation between the concept of capability and ” the schur’s theorem”. schur proved that the finiteness of g/z(g) implies the finiteness of γ2(g). a natural question which arises here is when the converse of the theorem of schur does hold? infinite extra special groups show that the converse of the schur’s theorem