Order of each element in a symmetry group.

Question

In abstract algebra, the order of an element in a cyclic group is defined as the smallest positive integer such that that element to the power of that integer yields the group's identity. However, why is the definition of order seem to be different for symmetry groups? For them, the definition of order is the number of times that it fits on to itself during a full rotation of 360 degrees. Or are those definitions the same?

Answer 1

The order of an element $a$ in any group $G$ is defined to be the least positive integer $n$ such that $a^n=e$, where $e$ is the identity element of the group.

For a cyclic group $C=\langle a\rangle$, the identity element is $e=a^0$.

For a symmetry group $S$ consisting some isometries $\mathbb{R}^2\to\mathbb{R}^2$, the identity element is the identity map $x\mapsto x$. It is easily seen that rotate by $2n\pi$ ($n\in\mathbb{Z}$) is the same as the identity map.