Please help with this question

Let G be the group of complex numbers 1,i,-1,-i under multiplication. Which of the following statements

are true about the homomorphisms of G into itself?

I. z-> z defines one such homomorphism, where z denotes the complex conjugate of z.

II. z-> z2 defines one such homomorphism.

III. For every such homomorphism, there is an integer k such that the homomorphism has the form z-> z^k .

(A) None (B) II only (C) I and II only (D) II and III only (E) I, II, and III

the correct answer is (E)

problem is with the third statement

III. For every such homomorphism, there is an integer k such that the homomorphism has the form z-> z^k .

thx

## gr0568 #46

f(x^n)=f(x)^n

and because x^n goes through all the elements of G, we see that the image of x completely characterizes the homomorphism.

In your exercise H=G and f(x)=x^k, so that for z=x^n

f(z)=f(x^n)=f(x)^n=x^nk=z^k.