There is no sharp transitivity on q6 when q is a type of Morley rank 2

Author:

Gropp Ursula

Abstract

In this paper we study transitive group actions.:G × XX, definable in an ω-stable theory, where G is a connected group and X a set of Morley rank 2, with respect to sharp transitivity on qα. Here q is the generic type of X (X is of degree 1 by Proposition (1)), for ordinals α > 0, qα is the αth power of q, i.e. (aβ)β < α, ⊨ qα iff (aβ)β < α is an independent sequence (in the sense of forking) of realizations of q, and G is defined to be sharply transitive on qα iff for all (aβ)β < α, (bβ)β < αqα there is one and only one gG with g.aβ = bβ for all β < α. The question studied here is: For which powers α of q are there group actions subject to the above conditions with G sharply transitive on qα?In §1 we will see that for group actions satisfying the above conditions, G can be sharply transitive only on finite powers of q. Moreover, if G is sharply transitive on qn for some n ≥ 2, then the action of the stabilizer Ga on a certain subset Y of X satisfies the conditions above with Ga being sharply transitive on qm−1, where q′ is the generic type of Y (Proposition (8)). Thus, there would be a complete answer to the question if one could find some n < ω such that there is no group action as above with G sharply transitive on qn, but for n – 1 there is. In this paper we prove that such n exists and that it is either 5 or 6. More precisely, in §2 we prove that there is no group action satisfying the above conditions with G sharply transitive on q6. This is the main result of this paper. It is not known to the author whether the same also holds for q5 instead of q6. However, it does not hold for q4, as is seen in §3. There we give an example provided from projective geometry, for a group action satisfying the above conditions with G sharply transitive on q4; for G we choose PGL(3, K) and for X the projective plane over K, where K is some algebraically closed field.

Publisher

Cambridge University Press (CUP)

Subject

Logic,Philosophy

Cited by 4 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Toward the recognition of PGLn via a high degree of generic transitivity;Communications in Algebra;2018-10-29

2. Recognizing PGL$_3$ via generic 4-transitivity;Journal of the European Mathematical Society;2018-05-07

3. RANK 3 BINGO;The Journal of Symbolic Logic;2016-12

4. GROUPS OF MORLEY RANK 4;The Journal of Symbolic Logic;2016-01-22

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