On the weight of finite groups

Abstract

For a finite group G, let W(G) denotes the set of the orders of the elements of G. In this paper we study jW(G)j and show that the cyclic group of order n has the maximum value of jW(G)j among all groups of the same order. Furthermore we study this notion in nilpotent and non-nilpotent groups and state some inequality for it. Among the result we show that the minimum value of jW(G)j is power of 2 or it pertains to a non-nilpotent group.