gr0568 #46

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

gr0568 #46

Postby HaRrY » Tue Mar 31, 2009 4:58 pm

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

eof
Posts: 9
Joined: Sun Oct 12, 2008 3:39 pm

Postby eof » Tue Mar 31, 2009 5:42 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.




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