# Range of sizes of orbits and number of orbits for some unspecified set $X$

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 $$X$$?

My attempt:

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?

For $$|X|=8$$, orbits can only have lengths 1,2 or 5.
You are correct for the $$|X|=11$$ case.