where {ΓkG}k≥1 is the lower central series (LCS) of the group, defined, inductively, by Γ1G = G and Γk+1G = (ΓkG,G), and the Lie bracket [x, y] is induced from the group commutator (x, y) = xyxy. Many properties of a group are reflected in properties of its associated graded Lie algebra. For instance, if G is finitely generated, then the abelian groups grk(G) are also finitely generated; their ...

We describe a criterion for showing that the equation s2+y2p = α3 has no non-trivial proper integer solutions for specific primes p > 7. This equation is a special case of the generalized Fermat equation xp + yq + zr = 0. The criterion is based on the method of Galois representations and modular forms together with an idea of Kraus for eliminating modular forms for specific p in the final stage...

‎The aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]‎. ‎In [1]‎, ‎Ungar and Chen showed that the algebra of the group $SL(2,mathbb C)$ naturally leads to the notion of gyrogroups ‎and gyrovector spaces for dealing with the Lorentz group and its ‎underlying hyperbolic geometry‎. ‎They defined the Chen addition and then Chen model of hyperbolic geomet...