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

1. ## 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.

2. 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 $\displaystyle a^2+b^2=|a+bi|^2$ and so $\displaystyle |(a+bi)(a'+b'i)|^2=|a+bi|^2|a'b'i|^2=1^2\cdot 1^2=1$

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

D) What's the canonical homomorphism?

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### show that the set g of all non zero complex is a group for multiplication of complex number

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