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On bad groups, bad fields, and pseudoplanes

Published online by Cambridge University Press:  12 March 2014

Ali Nesin
Ali Nesin*
Affiliation:
Department of Mathematics, University of California at Irvine, Irvine, California 92717
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Extract

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]

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 3 , September 1991 , pp. 915 - 931
Copyright
Copyright © Association for Symbolic Logic 1991

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References

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