In the variety of all groups, A. I. Mal’cev [5] proved in 1949 that the free product of two residually torsion-free nilpotent groups is again residually torsion-free nilpotent. This paper is motivated by the analogous question in the variety of metabelian groups: can we determine whether free metabelian products of residually torsion-free nilpotent metabelian groups are residually torsion-free ...

Recently, Macdonald et. al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in LOGSPACE. Here we follow their approach and show that all these problems are complete for the uniform circuit class TC0 – uniformly for all...

Let G be any group and let K(G) denote the multiplier Hopf algebra of complex functions with finite support in G. The product in K(G) is pointwise. The comultiplication on K(G) is defined with values in the multiplier algebra M(K(G)⊗K(G)) by the formula (∆(f))(p, q) = f(pq) for all f ∈ K(G) and p, q ∈ G. In this paper we consider multiplier Hopf algebras B (over C) such that there is an embeddi...

We establish the first super-logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group. In particular, for any constant c there are only finitely many finite p-groups of order pm with at most c ·m conjugacy classes. This answers a question of L. Pyber.

We consider an open problem on the stability of nonlinear nilpotent switched systems posed by Daniel Liberzon. Partial solutions to this problem were obtained as corollaries of global nice reachability results for nilpotent control systems. The global structure is crucial in establishing stability. We show that nice reachability analysis may be reduced to the reachability analysis of a specific...

We study complex product structures on nilpotent Lie algebras, establishing some of their main properties, and then we restrict ourselves to 6 dimensions, obtaining the classification of 6-dimensional nilpotent Lie algebras admitting such structures. We prove that any complex structure which forms part of a complex product structure on a 6-dimensional nilpotent Lie algebra must be nilpotent in ...

Some properties and characterizations for nilpotent matrices are established and in particular, a necessary and sufficient condition for an n × n nilpotent matrix to have the nilpotent index 2 and 3 is given. Mathematics Subject Classification: 15A33; 06D99

The K-groups, the range of trace on K0, and isomorphism classifications are obtained for simple infinite dimensional quotient C*-algebras of the group C*-algebras of six lattice subgroups, corresponding to each of the six non-isomorphic 5-dimensional connected, simply connected, nilpotent Lie groups. Connes’ non-commutative geometry involving cyclic cocycles and the Chern character play a key r...