A Generalization Of Sylow Theorem

We investigate the inverse of the Sylow's theorem: given prime number p, and a fixed positive integer k, does there always exists a finite group contains exactly kp+1 subgroups of size p? In the work, weprove under some conditions, there exists such a finite group. Using group extension theory and some number theory, we prove that the answer is yes for p=2. The dihedral group D_2k+1 of order 2k+l contains exactly 2k +1 subgroups of size 2. And whenkp +1 =(?)(n is an positive integer), the answer is also yes. The direct product of ncopies of cyclic groups of size p is an Abelian group of size pⁿ, and contains exactlykp+1 = (?) subgroups of size p. When kp+1 is a prime number, the answer is also yes.There exists a group of size p(kp +1) which and contains exactly kp +1 subgroups of size p, there are exactly (p-1)(kp+1) elements of order p in the group, and non unit elements from different such subgroups never commute. We also investigate the situation when a group contains exactly 7 subgroups of size 3, and discuss when do those elements commute. We conclude that there does not exist a group contains exactly 7 subgroups of size 3 such that1. elements from different such subgroups always commute, or2.there exists four subgroups, their elements mutually commute, and there are no other elements of order 3 commute.