# Actions of $Z_n$ and actions of $Z_{n-1}$

I am playing with some questions concerning connections between certain poset partitions and their linear extensions. This is not my usual playground, I just happened to stumble upon something.

When I was playing with these things, I came up with a very simple construction that I cannot find anywhere.I suspect that this construction is a minor particular case of something well known and much more general. It reminds of an equivariant map, but the group is not fixed here.

Let $n>2$, let $X$ be a finite set of cardinality $n$. Let $\oplus_n$ be an action of $Z_n$ on $X$ such that the action of $1$ is a cycle of length $n$. Pick $x\in X$ and let $\pi$ be a partition of the set $X$ such that $\{x,x\oplus_n 1\}$ is the only nonsingleton block of $\pi$.

In a very simple way, this gives rise to an action $\oplus_{n-1}$ of $Z_{n-1}$ on $\pi$ by ,,squeezing the action'' at $x$:

Put

• $\{x,x\oplus_n 1\}\oplus_{n-1} 1=\{x\oplus_n 2\}$
• $\{x\oplus_n(n-1)\}\oplus_{n-1} 1=\{x,x\oplus_n 1\}$
• $\{y\}\oplus_{n-1} 1=\{y\oplus_n 1\}$ for any $y\neq x\oplus_n(n-1)$.

It can be visualized in a simple way by a digraph construction: if we identify the action of $1$ on $X$ with an oriented cycle, this construction corresponds to a contraction of an edge.

Has anyone seen this construction before? Is there any name for it?

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Seems to me this could fail if n is not prime power. A faithful action corresponds to a set of cycles with LCM equal to n, not necessarily a single cycle. – Omer Apr 9 '11 at 14:55
You are right, of course. I have mixed up "regular" and "faithful". I have edited the question. – Gejza Jenča Apr 9 '11 at 21:05

I might be missing something, but it seems to me that there is not much going on in your construction. In fact, your original action of $Z_n$ on $X$ does nothing more than putting a cyclic ordering on your set $X$, and your complicated-looking construction creates a new cyclically ordered set $Y$ obtained from $X$ by removing one element (and retaining the ordering).