Suppose a group $G$ of order 10 acts on an infinite set $X$
Q1: What are all the possible sizes of the orbits of $G$?
Q2: How many orbits are there?
Now suppose $X$ has 8 elements. What are the possible sizes of orbits of $G$?
Now suppose $X$ has 11 elements. What’s the minimal number of $G$ orbits in $X$?
There could be some fixed point $x \in X$, but there could also exist a point that is moved to different points upon the action of different $g \in G$, so possible sizes of orbits would range from 1 to 10. It can also be the case that all points are fixed points under this action such that the number of orbits = cardinality of X or every $x \in X$ can be obtained by the action of some $g$ on a $y \in X$ such that there is only one orbit.
For $X$ having 8 elements, orbits can have sizes 1 to 8
For $X$ having 11 elements, there have to be at least two orbits since maximum size of one orbit is 10.
Is this correct?