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Math Help - cyclic groups

  1. #1
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    cyclic groups

    If n is a positive integer, then \sigma(n) will denote the cyclic group of order n.

    If gcd(m,n)=1, prove that \sigma(mn)\cong\sigma(m)\times\sigma(n).
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    Quote Originally Posted by deniselim17 View Post
    If n is a positive integer, then \sigma(n) will denote the cyclic group of order n.

    If gcd(m,n)=1, prove that \sigma(mn)\cong\sigma(m)\times\sigma(n).
    There is a known result which says if C is cyclic group of order n then C is isomorphic to \mathbb{Z}_n (or in your notation \sigma (n)). To show that \mathbb{Z}_m \times \mathbb{Z}_n is isomorphic to \mathbb{Z}_{nm} it is sufficient to show | \mathbb{Z}_m \times \mathbb{Z}_m | = nm (which is immediate) and also that \mathbb{Z}_m\times \mathbb{Z}_n is cyclic. Thus, we need to find a generator. Show that ([1]_m,[1]_n) has order nm and therefore it must generate the whole group. This will show the group is cyclic and complete the proof.
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