Abstract
Let G be a finite group. Recently various functions are defined related to the set of order elements of G and using these functions, some interesting criteria for solvability, nilpotency, supersolvability and etc., are obtained. In this paper, we continue this work and for a finite group G, we consider the function , where o(g) denotes the order of
. In this paper, we prove that if
or
, then G is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.