# Infinite group acts on a set such that an orbit of any length exists.

Give an example of an infinite group $G$ which acts on a set $S$ such that for each $n\in \mathbb{N}$ there is an orbit of this length.

Has anyone got an idea?

I've been trying something rotations in $\mathbb{C}$ of powers of matrices, something with eigenvalues, ... I haven't found any example yet..

Take $S$ to be the disjoint union of $\mathbb{Z}/n$ on which acts $G=\mathbb{Z}$.

• Could you elaborate further? $G=\mathbb{Z}$ and $S=\cup_n \mathbb{Z}/n$, and how does $G$ act on $S$ then? – dietervdf Jan 7 '18 at 21:06
• Tsemo, great answer as this is essentially the minimal one – Max Jan 7 '18 at 21:41
• Is this different from $\Bbb Q$ acting on $\Bbb Q/\Bbb Z$? – G Tony Jacobs Jan 7 '18 at 21:57

Take the subgroup of permutations of $\mathbb Z$ generated by a permutation that shifts each even number two to the right, and partitions the odd numbers into a cycle of length 1, a cycle of length 2, a cycle of length 3, ...

(The even numbers gives you an orbit of length $\infty$, which you may or may not need to provide).

You can definitely do this using the idea of complex rotations, as you were inclined to. The idea is that you want a linear transformation such that $e^{2i \pi/n}$ is an eigenvalue for every positive integer $n$, since the corresponding eigenvector has period $n$.

Let $S$ be the set of functions $f:\mathbb N\rightarrow \mathbb C$, where $\mathbb N$ is the set of positive integers. Let $\mathbb Z$ act on $S$ by saying that, $n\in\mathbb Z$ takes $f:\mathbb N\rightarrow\mathbb C$ to the function $$n*f(k)=f(k)\cdot e^{2ni\pi/k}.$$ In particular, $1\in\mathbb Z$ it acts on the $k^{th}$ component of the function $f$ by multiplication by multiplication by $e^{2i\pi/k}$. Obviously, a function $f$ which is non-zero only at $k$ has period $k$ under this action.

Recall the classification of group actions: every group action is a disjoint union of its orbits, which are transitive group actions. A transitive group action is isomorphic to $G/G_x$ where $G_x$ is the stabilizer of any element, and every subgroup appears in this way.

Hence the possible sizes of orbits of actions of $G$ are precisely the possible indices of subgroups of $G$. So your question reduces to:

Find an example of a group $G$ with a subgroup of every possible finite index.

The easiest example of such a group is $G = \mathbb{Z}$ but many others are possible.