We introduce the class of rank-metric geometric lattices and initiate study their structural properties. Rank-metric can be seen as $q$-analogues higher-weight Dowling lattices, defined by himself in 1971. fully characterize supersolvable compute characteristic polynomials. then concentrate on small whose polynomial we cannot compute, provide a formula for them under polynomiality assumption Wh...

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...

We introduce the notion of a Whitney dual graded poset. Two posets are duals to each other if (the absolute value of) their numbers first and second kind interchanged between two posets. define new types edge labelings which we call labelings. prove that every poset with labeling has dual. Moreover, show how explicitly construct using technique involving quotient As applications our main theore...

Abstract In this paper, we call a finite group G G an N L M NLM -group ( C NCM -group, respectively) if every non-normal cyclic subgroup of prime order or 4 (prime power order, in is contained maximal . Using the property -groups and -groups, give new necessary suffic...

Let X and Y be finite-type CW–complexes (X connected, Y simply connected), such that the rational cohomology ring of Y is a k–rescaling of the rational cohomology ring of X . Assume H∗(X,Q) is a Koszul algebra. Then, the homotopy Lie algebra π∗(ΩY ) ⊗ Q equals, up to k–rescaling, the graded rational Lie algebra associated to the lower central series of π1(X). If Y is a formal space, this equali...

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...