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Math Help - Automorphism

  1. #1
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    Automorphism

    1) What is Aut(G) and how do you show that the automorphism is isomorphic to a group i.e. Aut(C5) =~ C4, and what is Aut(C9) is isomorphic to which cyclic group, is it group C8?

    2) Describe Aut(Cn) explicitly for n = 24
    How do you show that Aut(C12) =~ C2 * C2 and Aut(C14) =~ C6 ?. Find the correspoding isomorphism for Aut(C24)

    Another question is: Does it mean that, C2 * C2 = C4
    C4 * C4 = C16
    C5 * C5 = C25
    and so on ... so in general Cn * Cn = C(n*n). I dont know

    Can any body help please? I'd appreciate your help.

    Many thanks
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    1) What is Aut(G) and how do you show that the automorphism is isomorphic to a group i.e. Aut(C5) =~ C4, and what is Aut(C9) is isomorphic to which cyclic group, is it group C8?

    2) Describe Aut(Cn) explicitly for n = 24
    How do you show that Aut(C12) =~ C2 * C2 and Aut(C14) =~ C6 ?. Find the correspoding isomorphism for Aut(C24)

    Another question is: Does it mean that, C2 * C2 = C4
    C4 * C4 = C16
    C5 * C5 = C25
    and so on ... so in general Cn * Cn = C(n*n). I dont know

    Can any body help please? I'd appreciate your help.

    Many thanks

    The automorphism of the cyclic group Z/nZ is (Z/nZ)^{\times}, which is of order \phi(n) (link)

    \phi(9)=6, thus Aut(C9) is not isomorphic to C(8).
    \phi(mn) = \phi(m)\phi(n), when gcd(m,n)=1.
    \phi(3*8)=\phi(3)\phi(8) = 2 *4 =8, so the order of Aut(C24) should be 8.

    A cyclic group is an abelian group by definition.

    C_{m} \times C_{n} = C_{mn}, when gcd(m,n)=1. Thus C_{4} \times C_{4} \neq C_{16}

    Note: Someone might give you a more thorough answer for this. This is my attempt to this problem.
    Last edited by aliceinwonderland; January 21st 2009 at 01:50 PM.
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