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By Ludwig Baumgartner

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M (as the I are maximal AnandPUlay 35 ideals). +bm) = rbi = 0, so bi= 0. Similarly bi= 0 Vi. Thus the subgroups BI direct sum which implies that RM(A) > m, contradiction. (3) I = is for all g E G. Proof: Let by (2) IiJk—Jn be the distinct conjugates of I. +hm for hi E G'. Thus there is an m such that the Ij fl Rm are pairwise distinct. But G acts transitively and definably on the Ij H Rm. As G is connected, there is only one. So (3) is proved. (4) R = K and is action on A (making A into a K-vector space) is definable.

There are types pe Si(G) of maximum U-rank; these are precisely the generic types of G. An important topic that we have not mentioned is the study of groups of finite Morley rank from the point of view that they should resemble algebraic groups over algebraically closed fields. As this is clearly false for abelian groups (consider Zpoo), some hypothesis of non-abelianness should be imposed. The major work was done by Cherlin [Ch], where he showed: (1) A Morley rank 1 group is abelian-by-finite (Actually this is due to Reineke; Cherlin noted that any (infinite) co-stable group has an infinite definable abelian subgroup).

Proof: Let by (2) IiJk—Jn be the distinct conjugates of I. +hm for hi E G'. Thus there is an m such that the Ij fl Rm are pairwise distinct. But G acts transitively and definably on the Ij H Rm. As G is connected, there is only one. So (3) is proved. (4) R = K and is action on A (making A into a K-vector space) is definable. Proof: As A is generated by the AS and I = IS Vg, it follows that 1 = 0. Thus "R = K" and the action of r E R on A is determined by its action on AI. More precisely: given that I = 0 it follows that the action of an element r E R on A is determined by its action on AI.

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