We characterize supersolvable lattices in terms of a certain modular type relation. McNamara and Thomas earlier characterized this class as those graded having maximal chain that consists left-modular elements. Our characterization replaces the condition gradedness with second modularity on

Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X, denoted DG(X), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we show that if G is nilpotent of class c or supersolvable of length c then G always acts with distinguishing number a...