Consider the set `S={1,-1,i ,-1}` for fourth roots of unity. Construct the composition table for multiplication on S and deduce its various properties.

Explanation

We may prepare the composition table for multiplication on A as given below:

Clearly, 1 is the identity element.

`(1)^{-1}=1,(-1)^{-1}=-1,(1)^{-1}=-1 and (-1)^{-1}=1.`

For associativity

`(1 times 1) times i=1 times i=i `

`1 times(1 times i)=1 times i=i`