On families of nilpotent subgroups and associated coset posets

Nilpotent groups with three conjugacy classes of non-normal subgroups

‎Let $G$ be a finite group and $nu(G)$ denote the number of conjugacy classes of non-normal subgroups of $G$‎. ‎In this paper‎, ‎all nilpotent groups $G$ with $nu(G)=3$ are classified‎.  

The Coset Equivalence Relation and Topologies on Subgroups

The paper studies the structure of the homogeneous space G/H, for G a Polish group and H < G a Borel, not necessarily closed subgroup of G, from the point of view of the theory of definable equivalence relations. It makes a connection between the complexity of the natural coset equivalence relation associated with G/H and polishability of H, that is, the possibility of introducing a Polish grou...

Optimal Subgroups and Applications to Nilpotent Elements

Let G be a reductive group acting on an affine variety X , and let x ∈ X be a point whose G-orbit is not closed. In this paper, we study G.R. Kempf’s optimal class of cocharacters of G attached to the point x; in particular, we consider how this optimality transfers to subgroups of G. Our main result says that if K is a G-completely reducible subgroup of G which fixes x, then the optimal class ...

Nilpotent Families of Endomorphisms of ...

Equidistribution of Dense Subgroups on Nilpotent Lie Groups

Let Γ be a dense subgroup of a simply connected nilpotent Lie group G generated by a finite symmetric set S. We consider the n-ball Sn for the word metric induced by S on Γ. We show that Sn (with uniform measure) becomes equidistributed on G with respect to the Haar measure as n tends to infinity. We also prove the analogous result for random walk averages.