Classification of groups of given order is a basic problem in Group Theory. By Jordon- Holder theorem, ”simple groups” can be considered as building blocks of finite groups. The classification of the finite simple groups is one of the most celebrated achievements of the last century.

On the other hand, due to Sylow theorems, finite ”p-groups” can also be considered as building blocks of finite groups. The classification of the finite p-groups of a given order up to isomorphism, is a near-impossible task. The classification is known up to order p7, where p is an odd prime. For p = 2, the classifications is known up to order 210. The difficulty in classification of p-groups of order pn for large n is due to non-appearance of any patterns in the known p-groups of smaller order.

However, it has been realized that, the ”classification of p-groups with a given property” could have patterns, and it is nowadays mostly implemented method in classification. We focus on p-groups according to the number of distinct conjugacy class sizes.

In this talk, we will briefly discuss why classification of finite p-groups of nilpotency class 2 is important and and some possible direction for this problem. We will then discuss finite p-groups G which have two distinct conjugacy class sizes, i.e., xG is a fixed positive integer, for each x ∈ G \ Z(G). If time permits, we will discuss the analogous study in the case of nilpotent Lie algebras.