چكيده فارسي :
Let m and n be positive integer numbers. In this note we study all nite groups that for every nite subsets M and N containing m and n elements, respectively, there exist x 2 M and y 2 N such that ⟨x; y⟩ is r-Kappe (call this condition Kr(m; n)). In fact we ned some bounds for m and n such that G 2 Kr(m; n) implies that G is Kappe and we nd a bound for order of G when G is not Kappe group in Kr(m; n) and r = 2; 3. Also we study all nite groups such that every two subsets M and N of G, containing m and n elements, there exist x 2 M and y 2 N, such that ⟨x⟩ is subnormal in ⟨x; y⟩, (call this condition S(m; n)), and we will ne some bounds for m and n such that all nite groups in this class are nilpotent. Also we nd a bound for order of G when G is a non-nilpotent nite S(m; n)-group.
