Is this a binary operation (second time)?

Question

I was reading about order of an element in a group and came across this example, but I am unable to understand or imagine how it could be a group. Can someone explain ?

Answer 1

The definition doesn't make much sense. What the author could have meant is a group $(G,+)$ where $G = \{0^\circ, 120^\circ, 240^\circ\}$ and the operation $+$ takes two elements of $G$ and adds them together so that, for example, $0^\circ + 120^\circ = 120^\circ$ and $240^\circ + 240^\circ = 120^\circ$ (since the angle $480^\circ$ is the same as $120^\circ$).

The identity element of the group would then be $0^\circ$ and the order of $0^\circ$ would thus be one. That of $120^\circ$ would be three since $$3\cdot 120^\circ=360^\circ\equiv 0^\circ,$$ and that of $240^\circ$ would also be three as $$3\cdot240^\circ=720^\circ\equiv 0^\circ.$$