# Let C* be the set of all nonzero complex numbers a+bi

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• Mar 17th 2010, 06:38 PM
snick
Let C* be the set of all nonzero complex numbers a+bi
A) Prove that C* is a group under multiplication

B) let H={a+bi € G|a^2 + b^2 = 1}. Prove that H is a subgroup of C*.

C) Prove that the set of nth roots of unity
Un is a subgroup of H

D) let G be the group of all real 2x2 matrices of the form (a,b, -b, a, where not both a and b are 0, under matrix multiplication. Show that C* and G are isomorphic.
• Mar 17th 2010, 06:45 PM
Drexel28
Quote:

Originally Posted by snick
A) Prove that C* is a group under multiplication

B) let H={a+bi € G|a^2 + b^2 = 1}. Prove that H is a subgroup of C*.

C) Prove that the set of nth roots of unity U
n is a subgroup of H

D) let G be the group of all real 2x2 matrices of the form (a,b, -b, a, where not both a and b are 0, under matrix multiplication. Show that C* and G are isomorphic.

A) This is pretty obvious. What trouble are you having?

B)Also obvious. Note though that $a^2+b^2=|a+bi|^2$ and so $|(a+bi)(a'+b'i)|^2=|a+bi|^2|a'b'i|^2=1^2\cdot 1^2=1$

C) It's easier to note that $\phi:I_n\to\mathbb{Z}_n$ given by $e^{\frac{2\pi i k}{n}}\mapsto k$ is an isomorphism.

D) What's the canonical homomorphism?