On c-capability and n-isoclinic families of a specific class of groups

Abstract

Let χ denote the class of all groups G such that (G) ∩ Z(G) = 1. In this paper, it is shown that the converse of Baer’s theorem holds for the groups in χ. Then we prove that the existence of the isomorphism between the center factors of the groups in χ suffices for those groups to be isoclinic. We also prove that the isoclinism coincides with the n-isoclinism in χ. Finally, we obtain a criterion for c-capability of finite groups in χ.