On two problems about order sequences of finite groups

The order sequence of a finite group G is a non-decreasing finite sequence formed of the element orders of G . Several properties of order sequences were studied by P. J. Cameron and H. K. Dey in a recent paper that concludes with a list of open problems. In this paper, we solve two of these problems by showing the following facts: 1) If there is a non-supersolvable/non-solvable group of order n , it is not always true that its order sequence is properly dominated by the order sequence of any supersolvable/solvable group of order n ; 2) the supersolvability of a finite group cannot be described by its order sequence.