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...

Let $G$ be a finite group and $\psi(G) = \sum_{g \in G} o(g)$, where $o(g)$ denotes the order of $g G$. First, we prove that if is $n$ >31\psi(C_n)/77$, $C_n$ cyclic $n$, then supersolvable. This proves conjecture M.~{T\u{a}rn\u{a}uceanu}. Moreover, M. Herzog, P. Longobardi Maj put forward following conjecture: If $H\leq G$, \leqslant \psi(H) |G:H|^2$. In sequel, by an example show this not sat...