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Math Help - Let C* be the set of all nonzero complex numbers a+bi

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    Question 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.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by snick View Post
    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?
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