Partitionen Liescher und algebraischer Gruppen

Algebraic groups with a distinguished conjugacy class

“…In the third section one also finds examples of algebraic Fischer groups with an abelian variety s the identity component. Moreover we determine there all non-affine algebraic Frobenius groups with connected kernel; this corrects a claim in [12] (theorem 9.7) and completes the classification of algebraic Frobenius groups with connected kernel, since the affine case has been treated by Hertzig [5].…”

“…The examples we have exhibited in this section show that by the theorem 9.7 in [12] not all non-affine algebraic groups admitting a non-trivial partition into subgroups are classified s had been claimed there. For algebraic Frobenius groups this gap (which arises from a wrong calculation in lemma 9.4) is closed by our theorem 3.2.…”

“…For algebraic Frobenius groups this gap (which arises from a wrong calculation in lemma 9.4) is closed by our theorem 3.2. If G is a non-affine algebraic group which is not a Frobenius group, but which admits a non-trivial partition φ into subgroups then the connected component G° of G is contained in a member U of φ (this follows from the correct part of theorem 9.7 in [12] since the connected component of G admits only the trivial partition). If we take an element xe G of prime order q such that <x> n U = l, then the group K = G° <*> has a non-trivial partition since any element y e K\U has also order q (see theorem 9.7 in [12]).…”

“…Let L be the maximal connected affine algebraic subgroup of G°. Then by [10] the group L is nilpotent since the affine algebraic group L <x> has a non-trivial partition with L member of such partition (see [10] or corollary 9.17 in [12]). Hence G° is a nilpotent group.…”

Stable kinematic planes and sharply 2-transitive Lie groups

“…As a by-product we obtain a classification of stable kinematic planes. The paper is organized as follows: After a very short introduction to stable planes (for a detailed survey see Grundhöfer and Löwen [10]), we summarize some results of Plaumann-Strambach [24] concerning locally compact groups with partitions. As a new result, we prove that a partition P of a connected (Lie) group G is uniquely determined by the induced partition T e P = {T e X | X ∈ P} of its Lie algebra T e G, see Thm.…”