The Structure Of Compact Groups
This chapter investigates the structure of finite dimensional compact groups and the structure of compact transformation groups acting on finite dimensional spaces. The groups of the series are not required to be Lie groups themselves, but it is asked that each group of the sequence should be the homomorphic image of the next under a homo-morphism whose kernel is a Lie group. The main business of the present section is to show that an effective and transitive compact transformation group acting on a finite dimensional space is itself finite dimensional. Moreover, it will be shown that if this 0-dimensional space is finite then the transformation group is necessarily a Lie group. In particular, the transformation group is a Lie group if the space on which it acts is locally connected. The concept of the limit of a transfinite sequence is introduced for the sake of completeness, though it will be employed only in the construction of examples.