Group Action/Examples/Cyclic Group on Polygon/Mistake

Source Work

2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.):

action 3.

Mistake

For example, the $n$-element ${}^*$cyclic group whose elements are $e, a, a^2, \ldots, a^{n - 1}$ acts on the vertices of a ${}^*$regular polygon by the map for which $e x$ is $x$ for each vertex $x$, and $a^k x$ is the vertex obtained when $x$ is rotated through $2 \pi k / n$ radians about the center of the polygon.

Correction

It is not $x$ which is rotated, it is the whole polygon.

Hence this would be better worded as:

For example, the $n$-element ${}^*$cyclic group whose elements are $e, a, a^2, \ldots, a^{n - 1}$ acts on the vertices of a ${}^*$regular polygon by the map for which $e x$ is $x$ for each vertex $x$, and $a^k x$ is the vertex obtained when the polygon is rotated through $2 \pi k / n$ radians about the center of the polygon.

Sources

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