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Math Help - Additive order of elements in Euclidean Rings

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    Additive order of elements in Euclidean Rings

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    Last edited by abstracto; March 3rd 2010 at 08:17 AM.
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  2. #2
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    Quote Originally Posted by abstracto View Post
    I'm stumped by this problem. Any help would be greatly appreciated

    Show that the additive order of a Euclidean ring is either prime, infinite, or one, where the additive order of g in (R,+,*) is the order of g in group (R,+)

    Also, if anyone can help me find all the group homomorphisms from the cyclic group Z12 to alternating group A5, that would be awesome too.

    Thanks in advance!
    you don't need R to be Euclidean. the claim is true in any domain because in every domain all non-zero elements have the same additive order, which is the charateristic of the domain.

    for the second question just note that a group homomorphism f is completely determined by f(1)=\sigma, where \sigma is any element of A_5 whose order is a divisor of 12. so we must have

    o(\sigma) \in \{1,2,3,4,6,12 \}. but A_5 has no element of order 4, 6 or 12 and it has 20 elements of order 3 and 15 elements of order 2. so there are 36 group homomorphisms from \mathbb{Z}_{12} to A_5.
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