I spent some time revisiting group actions this week. I was hoping to get someone to verify a seemingly straightforward claim. I also had a thought on how "uniform" an action is on a space.
Re-reading my algebra text, it dawned on me that the reason we care about transitive group actions is because any group action can be decomposed into a collection of transitive actions. I was hoping someone could verify this for me:
To be more precise, let $G$ be a group and $A$ be a set it acts on. The action defines a partition of $A$ into orbits, which I will denote $\{A_i\}$. If we realize the action as a group homomorphism $G \rightarrow \text{Aut}(A)$, then we can restrict our action to any orbit as an action $G \rightarrow \text{Aut}(A_i)$, since for any $a \in A_i$, we have $ga \in A_i$.
This should be somewhat analogous to restricting a function on its domain or a linear operator on an invariant subspace.
Going the other way, given a partition $\{A_i\}$ of a set $A$, to define a group action $G \rightarrow \text{Aut}A$, it suffices to specify a transitive group action of $G$ on each $A_i$.
Assuming I haven't made a mistake up to this point, I would wonder how well this would work when $A$ is taken to be something more sophisticated than a mere set, say, a topological space, whose partitions and automorphisms are more limited.
Finally, I was also wondering, (again assuming that my claim above holds), if there is some well-established notion of "uniformity" of a group action on a space.
Consider a disjoint union of circles. Let $\mathbb{R}$ act on this space in the following way: for points $x$ in the "top" circle, $r \in \mathbb{R}$ rotates $x$ by $r$ radians, and for points $x$ in the "bottom" circle, $r \in \mathbb{R}$ rotates $x$ by $r$ degrees instead.
This would be an example of a "non-uniform" action, since the transitive action on the top circle is not "the same" as the transitive action on the bottom circle.
Is this a standard notion in group theory? I'm sure someone has studied this and has given a name to it.
It seems that a group action should be called uniform if any automorphism $\phi \colon A \rightarrow A$ (thinking of A here as a topological space) doubles as a $G$-set isomorphism between $\rho \colon G \rightarrow \text{Aut}(A)$ and $\rho^\prime \colon G \rightarrow \text{Aut}(A)$ defined as $\rho^\prime(g, x) := \rho(x, \phi(x))$.
Clearly, these thoughts are only half-baked, but I figured I would fish to see if these are standard notions in group theory. Thanks.