In the multiplicative group G={1,w,w^2}, the inverse of w^2 is:a) 1 b) w^2 c) w d)None of these

Find the inverse for the element `w^2` in the multiplicative group `{1,w,w^2}` :

You could build a multiplication table:

        1   w   `w^2`
       ------------

1 |    1   w   `w^2`

w |    w  `w^2`   1

`w^2` | `w^2 `   1   w

Note that `w*w^2=w^3=1` and `w^2*w^2=w^4=w^3*w=1*w=w`

The inverse of `w^2` is w.

** This is a group. Every element has an inverse for the operation defining the group, and the inverse must be an element of the group, so the answer cannot be (d). Assuming 1 is the identity element, 1 cannot be the inverse of a non-identity element so the answer cannot be (a). Multiplicative groups are abelian so `w*w^2=w^2*w` .

This is a cyclic group (order is prime=3), so we have `w^0=1,w^1=w,w^2=w^2,w^3=w^0=1` etc... (The exponents are reduced modulo 3.)