The purpose of this paper i s to give a relati on b etween the notion of the commutativity degr ee of a fini te gr oup G (denoted by d(G)) and that o f i s o cl i n i s m b etween G and an extr a s p eci al p - gr oup, wher e p is the smallest pr ime numb er dividing | G | .Moreover, s o m e i m pr ovem ents of the r es ul ts on the r el ati ve com mu tati vi ty degr ee and r el ati ve n th ni l p otency degr ee of a s ubgr oup of fini te gr oups gi ven i n [ A . Er f ani an, R . R ezaei and P. Les cot, O n the r el ati ve com mutati v i ty d egr ee of a s ubgr oup of a fini te gr oup, Comm. Algebra 35 (2007) 4183–4197] ar e a l s o s tated i n thi s p ap er .

Keywords