Group Action/Examples/Cyclic Group on Polygon

Example of Group Action

Consider the cyclic group $C_n$ defined as $\gen g$ whose identity is $e$.

Let $P_n$ be a regular $n$-sided polygon.

Then $C_n$ acts on on $P_n$ by the mapping for which, for each vertex $x$ of $P_n$:

$e x = x$

$g^k x$ is the vertex obtained when $P_n$ is rotated through $\dfrac {2 \pi k} n$ radians about the center of $P_n$.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): action: 3. (of a group on a set)