Solution:

Let four roots of unity be $G=\{1,-1, i,-i\}$.

The composition table of G is -

$ \begin{array}{|c|c|c|c|c|} \hline * & 1 & -1 & i & -i \\\\ \hline 1 & 1 & -1 & i & -i \\\\ \hline-1 & -1 & 1 & -i & i \\\\ \hline i & i & -i & -1 & 1 \\\\ \hline-i & -i & i & -1 & i \\\\ \hline \end{array}\\ $

i) Closure property;-

Since all the entries of the composition table are the elements of given set, the set G is closed under multiplication.

ii) Associativity : -

The element of G are complex numbers and we know that multiplication of complex numbers is associative.

iii) Identity_:-

Here, 1 is the identity element.

iv) Inverse:-

From the composition table, we see that inverse elements of 1,-1, i, -i are 1 , -1, -i, i respectively.

v) Commutativity:-

The corresponding rows and columns of the table are identical. Therefore, the binary operation is commutative. Hence, $(G, *)$ is an finite abclian group.