Transitivity

Abstract

Most of group theory, and in particular the theory of group actions, is about symmetries. A natural question to ask therefore is whether two points can look the same. Translated into our language of group actions, this question can be stated as follows. Given two points α, β in a G-space Ω, can one find an element g ∈ G such that α^g = β? In this chapter, we formalise this question and discuss the immediate consequences if the answer to the above question is in the affirmative. For further reading, the reader is again referred to Wielandt (1960, 1964).