Multiplication on three elements in a binary operation

Question

how that the set G={ͳ, , ^2} form an abelian group under the operation of ordinary multiplication where ͳ, , ^2 are cube roots of units.

here can you please explain how to get the multiplication of these three elements

Answer 1

Not sure what that first symbol is, but it must be the identity. If $\nu$ were the identity, $\nu^2$ would be too, so they would not be distinct elements. So $\nu, \nu^2$ are not $1$.

You can use the fact that each element in $G$ cubes to $1$ to figure out the multiplication table. E.g.,

\begin{align*} \nu^2 \cdot \nu &=\nu^3=1\\ \nu^2 \cdot \nu^2 &= \nu^3\cdot \nu = 1\cdot \nu = \nu \end{align*}

and so on.