Generalized Kaloujnine groups, uniseriality and height of automorphisms

We show that the Lie action of the Kaloujnine group K(p, n) on the vector space (Fp)(pn) is uniserial. Using some Radon transform techniques we derive a formula for the height of the elements in K(p, n). A generalization of the Kaloujnine groups is introduced by considering automorphisms of a spherically homogeneous tree. We observe that uniseriality fails to hold for these groups and determine their lower central series; finally we discuss in detail Kaloujnine's description of the characteristic subgroups in terms of the (normal) "parallelotopic" subgroups.