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Math Help - abstract algebra: cyclic!

  1. #1
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    Question abstract algebra: cyclic!

    can someone please help? i'm studying for an exam and i'm stuck on this problem.

    Prove that U(2^n) (n>less than 3) is not cyclic.

    thanks so much
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  2. #2
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    Quote Originally Posted by dlin3 View Post
    can someone please help? i'm studying for an exam and i'm stuck on this problem.

    Prove that U(2^n) (n>less than 3) is not cyclic.

    thanks so much
    1)Prove that 5^{2^{n-3}} \equiv 1 + 2^{n-1} ~ (2^n) for n\geq 3 by induction.
    2)Now show that 5^{2^{n-2}} \equiv 1 ~ (2^n) and 5^{2^{n-3}} \not \equiv 1 ~ (2^n).
    3)Show that \{ \pm 5^k | 0\leq k < 2^{n-1} \} = U(2^n).
    4)Therefore no element has order \phi (2^n) in U(2^n).
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  3. #3
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    How did you get 5^(2n-3) = 1 + 2^(n-1) * (2n)?
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  4. #4
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    Quote Originally Posted by dlin3 View Post
    How did you get 5^(2n-3) = 1 + 2^(n-1) * (2n)?
    Use what I said above. Use induction.
    And the fact if a\equiv b(\bmod p^k) then a^p \equiv b^p (\bmod p^{k+1}) (where p is prime, in this case two).
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