Which of the following functions are homomorphisms?

Part A:

Given defined by



but



Since, f(xy) ≠ f(x)f(y)

Therefore, the function is not a homomorphism.

Part B:

Given defined by

Note that in , -1 = 1 and f(0) = 0 and f(1) = -1 = 1, so we can also use the formular



and



Therefore, the function is a homomorphism.

Part C:

Given , defined by



Since, f(x+y) ≠ f(x) + f(y), therefore, the function is not a homomorphism.

Part D:

Given , defined by



but



Since, h(ab) ≠ h(a)h(b), therefore, the funtion is not a homomorphism.

Part E:

Given , defined by , where denotes the lass of the integer in .

Then, for any , we have



and



Therefore, the function is a homomorphism.

The answer to the question is d