In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of chief factors for Polish groups. In the development of this theory, we prove a version of the Schreier refinement theorem. We also prove a trichotomy for the structure of topologically characteristically simple Polish groups.

The development of the theory of chief factors requires two independently interesting lines of study. First we consider injective, continuous homomorphisms with dense normal image. We show such maps admit a canonical factorisation via a semidirect product, and as a consequence, these maps preserve topological simplicity up to abelian error. We then define two generalisations of direct products and use these to isolate a notion of semisimplicity for Polish groups.

在有限群论中，主因子在结构理论中扮演着重要的角色，并且是众所周知的。我们在这里发展了波兰群体的主要因素理论。在该理论的发展过程中，我们证明了 Schreier 改进定理的一个版本。我们还证明了拓扑特征简单波兰群的结构的三分法。

主因素理论的发展需要两条独立有趣的研究方向。首先，我们考虑具有密集法线图像的单射、连续同态。我们展示了这样的映射通过半直接乘积承认规范分解，因此，这些映射保持拓扑简单性直至阿贝尔误差。然后，我们定义了直接产品的两个推广，并使用它们来隔离波兰群的半简单性概念。