0568 Q46
Posted: Thu Jan 19, 2012 10:46 pm
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-> z^2 defines one such homomorphism.
III. For every such homomorphism, there is an integer k such that the homomorphism has the form z ->z^k .
Hi, may I know how would we prove assertion (III)? Is it by considering cases? My group theory is not very strong, but I learnt that homomorphism must map 1 to 1, so that leaves us 3!=6 cases to consider?
Thanks a lot.
For (I), f(ab)=(ab)*=a*b*=f(a)f(b)
For (II), f(ab)=(ab)^2=(a^2)(b^2)=f(a)f(b).
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-> z^2 defines one such homomorphism.
III. For every such homomorphism, there is an integer k such that the homomorphism has the form z ->z^k .
Hi, may I know how would we prove assertion (III)? Is it by considering cases? My group theory is not very strong, but I learnt that homomorphism must map 1 to 1, so that leaves us 3!=6 cases to consider?
Thanks a lot.
For (I), f(ab)=(ab)*=a*b*=f(a)f(b)
For (II), f(ab)=(ab)^2=(a^2)(b^2)=f(a)f(b).