A Generic Identification Theorem for Groups of Finite Morley Rank

Abstract

The paper contains a final identification theorem for the 'generic' K * -groups of finite Morley rank.

A Generic Identification Theorem for L∗-groups of finite Morley rank

“…Then x acts on the set Σ , since x ∈ N(T ) N(D), by [15,Lemma 3.8]. If L ∈ Σ then L ∩ L x contains T ∩ L. If L = L x then L ∩ L x Z(L) has order at most two, a contradiction.…”

“…The next result occurs as a hypothesis in the version of Niles' theorem given in [15] and quoted above. It presents no difficulty in the K * context, but does require attention in the L * context, being a point where degenerate sections could intervene strongly, in principle.…”

“…The proof will follow the line of [15]. We retain the hypotheses and notation of this theorem throughout the present section.…”

“…In the K * -case, the analysis of groups of even type was undertaken in [2][3][4]7,25], and this material is being adapted to the L * -case, see in particular [9][10][11][12] for the first part of this adaptation, with more to come. Our recognition theorem has also been given in the K * -case [15], and must be adapted to the L * setting. We also enlarge on some earlier arguments that were given rather sketchily in earlier accounts, and correct some inaccuracies.…”

“…Our main theorem is an adaptation of [15] to the context of L * -groups of even type. It concerns simple L * -groups of even type (see Section 1.4 for general terminology).…”