Group Action/Examples/Cyclic Group on Polygon
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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)