Nilpotent Singer groups

Let $N$ be a nilpotent group normal in a group $G$. Suppose that $G$ acts transitively upon the points of a finite non-Desarguesian projective plane $\mathcal{P}$. We prove that, if $\mathcal{P}$ has square order, then $N$ must act semi-regularly on $\mathcal{P}$. In addition we prove that if a finite non-Desarguesian projective plane $\mathcal{P}$ admits more than one nilpotent group which is regular on the points of $\mathcal{P}$ then $\mathcal{P}$ has non-square order and the automorphism group of $\mathcal{P}$ has odd order

Homotopy nilpotent groups

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $\pi_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $\Omega F$ have values in homotopy n-nilpotent groups.Comment: 16 pages, uses xy-pic, improved exposition, submitte

Generalized Analogs of the Heisenberg Uncertainty Inequality

We investigate locally compact topological groups for which a generalized analogue of Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $\mathbb{R}^n \times K$ (where $K$ is a separable unimodular locally compact group of type I), Euclidean Motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie groups, $2$-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups

On groups covered by locally nilpotent subgroups

Let N be the class of pronilpotent groups, or the class of locally nilpotent profinite groups, or the class of strongly locally nilpotent profinite groups. It is proved that a profinite group G is finite-by-N if and only if G is covered by countably many N-subgroups. The commutator subgroup G\ue2\u80\ub2is finite-by-N if and only if the set of all commutators in G is covered by countably many N-subgroups. Here, a group is strongly locally nilpotent if it generates a locally nilpotent variety of groups. According to Zelmanov, a locally nilpotent group is strongly locally nilpotent if and only if it is n-Engel for some positive n