论文标题
约旦 - 切瓦利的分解超出代数群体
A Jordan-Chevalley decomposition beyond algebraic groups
论文作者
论文摘要
我们证明了可定义的组在O最低结构中的分解，从而概括了线性代数组的Jordan-Chevalley分解。因此，任何可定义的线性G是其最大正常可定义的无扭转亚组N（g）和可定义的亚组P的半程乘积，其独特性是共轭性，绝对同构对半ger型组。 在此过程中，我们在任意O最低结构中建立了古典群体的另外两个基本分解：1）LEVI分解和2）依赖于Frattini论点对O-最低环境的概括的断开组的关键分解。在O最低结构中，与P组一起，0组起着至关重要的作用。我们给出了这两个类别的表征，并表明可确定的P组是可以解决的，例如有限的P组，但它们不一定是nilpotent。此外，我们证明可定义的p组（p = 0或p prime）是由扭转元素确定产生的，并且在确定连接的组中，0-Sylow子组与每个P priper的p-Sylow子组一致。
We prove a decomposition of definable groups in o-minimal structures generalizing the Jordan-Chevalley decomposition of linear algebraic groups. It follows that any definable linear group G is a semidirect product of its maximal normal definable torsion-free subgroup N(G) and a definable subgroup P, unique up to conjugacy, definably isomorphic to a semialgebraic group. Along the way, we establish two other fundamental decompositions of classical groups in arbitrary o-minimal structures: 1) a Levi decomposition and 2) a key decomposition of disconnected groups, relying on a generalization of Frattini's argument to the o-minimal setting. In o-minimal structures, together with p-groups, 0-groups play a crucial role. We give a characterization of both classes and show that definable p-groups are solvable, like finite p-groups, but they are not necessarily nilpotent. Furthermore, we prove that definable p-groups (p=0 or p prime) are definably generated by torsion elements and, in definably connected groups, 0-Sylow subgroups coincide with p-Sylow subgroups for each p prime.
