On Bad Groups, Bad Fields, and Pseudoplanes

Abstract

Cherlin introduced the concept of bad groups (of finite Morley rank) in [Ch1]. The existence of such groups is an open question. If they exist, they will contradict the Cherlin-Zil'ber conjecture that states that an infinite simple group of finite Morley rank is a Chevalley group over an algebraically closed field. In this paper, we prove that bad groups of finite Morley rank 3 act on a natural geometry Γ (namely on a special pseudoplane; see Corollary 20) sharply flag-transitively.We show that Γ is not very far from being a projective plane and when it is so rk(Γ) = 2 and Γ is not Desarguesian (Theorem 2). Baldwin [Ba] recently discovered non-Desarguesian projective planes of Morley rank 2. This discovery, together with this paper, makes the existence of bad groups (also of bad fields) more plausible. A bad field is a pair ( K, A ) of finite Morley rank, where K is an algebraically closed field, A ≠ K * and A is infinite. There existence is also unknown. In this paper, we define the concept of a sharp-field as a pair ( K, A ), where K is a field, A K *and 1. K = A − A , 2. If a + b − 1 ∈ A , a ∈ A , b ∈ A , then either a = 1 or b = 1. If K is finite this is equivalent to 1 and 2.′ ∣ K ∣ = ∣ A ∣ 2 ∣ A ∣ + 1. Finite sharp-fields are special cases of difference sets [De]