We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group AΓ with connected defining graph. We use this to prove that the divergence of AΓ is linear if Γ is a join and quadratic otherwise. As an application, we give a complete description of the cut points in any asymptotic cone of AΓ. We also show that every non-abelia...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

The minimal faithful permutation degree n(G) of a finite group G is the least positive integer n such that G is isomorphic to a subgroup of the symmetric group Sn. Let AT be a normal subgroup of a finite group G. We prove that n(G/N) $C /i(G) if G/N has no nontrivial Abelian normal subgroup. There is an as yet unproved conjecture that the same conclusion holds if G/N is Abelian. We investigate ...

We suggest a new (dynamical) Abelian projection of the lattice QCD. It contains no gauge condition imposed on gauge fields so that Gribov copying is avoided. Configurations of gauge fields that turn into monopoles in the Abelian projection can be classified in a gauge invariant way. In the continuum limit, the theory respects the Lorentz invariance. A similar dynamical reduction of the gauge sy...

let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

The algebraic conditions that specific gaugedG/H-WZWmodel have to satisfy in order to give rise to Non-Abelian Toda models with non singular metric with or without torsion are found. The classical algebras of symmetries corresponding to grade one rank 2 and 3 singular NA-Toda models are derived. The proliferation of different two dimensional G0-Toda theories ([2] [10] ) addresses the question a...

In abstract algebra courses, teachers are often confronted with the task of drawing subgroup lattices. For purposes of instruction, it is usually desirable that these lattices be planar graphs (with no crossings). We present a characterization of abelian groups with this property. We also resolve the following problem in the abelian case: if the subgroup lattice is required to be drawn hierarch...

We prove that every finite abelian group G occurs as a subgroup of the class infinitely many real cyclotomic fields.

Let p be an odd prime. We construct a non-abelian extension Γ of S1 by Z/p × Z/p, and prove that any finite subgroup of Γ acts freely and smoothly on S2p−1 × S2p−1. In particular, for each odd prime p we obtain free smooth actions of infinitely many non-metacyclic rank two p-groups on S2p−1 × S2p−1. These results arise from a general approach to the existence problem for finite group actions on...

A measure preserving action of a countably in nite group Γ is called totally ergodic if every in nite subgroup of Γ acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of Γ is totally ergodic then there exists a nite normal subgroup N of Γ such that the stabilizer of almost every point is equal to N. Surprisingly the proof r...