Answering a question raised by Terwijn, we give a lattice-theoretic characterization of the intervals of the Muchnik lattice as a subset of the distributive lattices, valid for all intervals satisfying a cardinality constraint on the size of antichains.

It is known from work by H.Abels and P.Abramenko that for a classical Fqgroup G of rank n the arithmetic lattice G(Fq[t]) of Fq[t]-points is of type Fn−1 provided that q is large enough. We show that the statement is true without any assumption on q and for any isotropic, absolutely almost simple group G defined over Fq. Let k be a global function field and let G be a connected, noncommutative,...

We briefly discuss the history of lattices and cryptography during the last fifteen years. A lattice is a discrete subgroup of R or equivalently the set L

We consider several subgroup-related algorithmic questions in groups, modeled after the classic computational lattice problems, and study their computational complexity. We find polynomial time solutions to problems like finding a subgroup element closest to a given group element, or finding a shortest non-trivial element of a subgroup in the case of nilpotent groups, and a large class of surfa...

Complex hyperbolic triangle groups, originally studied by Mostow in building the first nonarithmetic lattices in PU(2, 1), are a natural generalization of the classical triangle groups. A theorem of Takeuchi states that there are only finitely many Fuchsian triangle groups that determine an arithmetic lattice in PSL2(R), so triangle groups are generically nonarithmetic. We prove similar finiten...

let $g$ be a finite group‎. ‎a subgroup‎ ‎$h$ of $g$ is called an $mathcal h $ -subgroup in‎ ‎$g$ if $n_g (h)cap h^gleq h$ for all $gin‎ ‎g$. a subgroup $h$ of $g$ is called a weakly‎ $mathcal h^ast $-subgroup in $g$ if there exists a‎ ‎subgroup $k$ of $g$ such that $g=hk$ and $hcap‎ ‎k$ is an $mathcal h$-subgroup in $g$. we‎ ‎investigate the structure of the finite group $g$ under the‎ ‎assump...

Abstract Given a finite group G , we denote by $L(G)$ the subgroup lattice of and ${\cal CD}(G)$ Chermak–Delgado . In this note, determine groups such that $|{\cal CD}(G)|=|L(G)|-k$ for $k=1,2$