关于有限莫利秩的无限幂零群的性质探讨

Study of an Infinite Nilpotent Group of Finite Morley Rank Properties

用模型论与代数方法相结合来研究无限群的性质,因为任何一个群G同构于在代数闭域上的代数群,则TH(G)理论是ω稳定的,而有限莫利秩无限群类似于在代数闭域上的代数群;已知有限莫利秩的无限群具有降链条件,利用降链条件证明了连通有限莫利秩的无限幂零群G的几个性质,群G的中心群Z(G)包含任何有限正规子群;中心群Z(G)是无限的;对任何群G的真子群H,都有[N(H):H]是无限的。

This paper researched the character of infinite group by using model theory combined with algebraic method,any finite group G was isomorphic to an algebraic group which is in the algebraic closed field,this meant TH（ G） theory was stable,yet the group of finite Morley rank was similar to the algebraic group which is in the algebraic closed field. It was known that the infinite group of the group of finite Morley rank had the descending chain condition,the characters of infinite nilpotent group G which connected the group of finite Morley rank was proved by descending chain condition,the central group Z（ G） of group G contained any finite normal subgroup,central group Z（ G） was infinite,[N（ H） ： H] was infinite for any proper subgroup H of group G.