## gr0568 #46

Forum for the GRE subject test in mathematics.
HaRrY
Posts: 1
Joined: Tue Mar 31, 2009 1:02 pm

### gr0568 #46

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

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

eof
Posts: 9
Joined: Sun Oct 12, 2008 3:39 pm
The group is cyclic. In general if G and H are groups and f:G->H is a homomorphism. Then if x is a generator for G, we have

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.