finite groups with the kappe property

Let m and n be positive integer numbers. in this note we studyall nite groups that for every nite subsets m and n containing m andn elements, respectively, there exist x 2 m and y 2 n such that ⟨x; y⟩ isr-kappe (call this condition kr(m; n)). in fact we ned some bounds for mand n such that g 2 kr(m; n) implies that g is kappe and we nd a boundfor order of g when g is not kappe group in kr(m; n) and r = 2; 3. also westudy all nite groups such that every two subsets m and n of g, containingm and n elements, there exist x 2 m and y 2 n, such that ⟨x⟩ is subnormalin ⟨x; y⟩, (call this condition s(m; n)), and we will ne some bounds for mand n such that all nite groups in this class are nilpotent. also we nd abound for order of g when g is a non-nilpotent nite s(m; n)-group.