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Math Help - Cyclic groups modulo

  1. #1
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    Cyclic groups modulo

    Hi all,

    Brief intro to what im talking about....

    If n \geq 2 is an integer, then Zn* denotes the set of invertible elements in the ring Zn. That is, it denotes the numbers in {1,2....n-1} which are coprime to n. The set Zn* is a group under multiplication modulo n.

    Does anyone know how to determine whether or not these groups are cyclic?: Z8*, Z9*, Z10*, Z12*

    Maybe someone can walk me through the first couple and i can try the others for myself!

    Thanks!
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  2. #2
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    Quote Originally Posted by sirellwood View Post
    Hi all,

    Brief intro to what im talking about....

    If n \geq 2 is an integer, then Zn* denotes the set of invertible elements in the ring Zn. That is, it denotes the numbers in {1,2....n-1} which are coprime to n. The set Zn* is a group under multiplication modulo n.

    Does anyone know how to determine whether or not these groups are cyclic?: Z8*, Z9*, Z10*, Z12*

    Maybe someone can walk me through the first couple and i can try the others for myself!

    Thanks!

    This is standard stuff you can find in any decent book in group theory or general algebra. Read the folowing, too:
    Multiplicative group of integers modulo n - Wikipedia, the free encyclopedia

    Tonio
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  3. #3
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    here is how you would proceed for U(Z8): the elements co-prime to 8 are the odd integers {1,3,5,7} (since 8 is a power of 2). this is a group of order 4, for it to be cyclic, you would need an element of order 4. it suffices to check 3,5 and 7. 3^2 = 1, 5^2 = 1, and 7^2 = 1, so U(Z8) is not cyclic.

    U(Z9) is even easier: it has 6 elements {1,2,4,5,7,8}. we know that the multiplication is commutative, and any abelian group of order 6 is cyclic.
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