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Math Help - Group Homomorphisms

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    Super Member Bernhard's Avatar
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    Group Homomorphisms

    I am trying to understand homomorphisms from  Z_n to  Z_k and despite some help from earlier posts am still struggling.

    I am thus reading Beachy and Blair: Abstract Algebra on group homomorphisms and am trying to follow Example 3.7.7 on page 156 which seeks to determine all homomorphisms from  Z_n to  Z_k (see relevant page attached). I can follow this example but am having trouble with a particular step.

    I will present their argument down to the problematic step - and I will make the example more concrete by seeking to determine all homomorphisms  \phi :  Z_6 \rightarrow  Z_{10}

    In terms of my particular example the argument of Beachy and Blair is as follows:

    ================================================== ====
    Consider  \phi :  Z_6 \rightarrow  Z_{10}

    Any such homomorphism is completely determined by  \phi ( [1]_6), and this must be an element [m]_{10} of Z_{10} whose order is a divisor of 6.

    In an abelian group with the operation denoted additively we have that if o(a) | n if and only if n.a = 0.

    Applying this result to [m]_{10} in Z_{10} we have o( [m]_{10}) | 6 if and only if n. [m]_{10} = [0]_{10} which happens if and only if 10 | 6m (Beachy and Blair ask us to compare with Exercise 11 of Section 2.1 - see other attached sheet)

    etc etc - see attached sheet
    ================================================== ==

    My question is how does the step that concludes " which happens if and only if 10 | 6m" follow ie how does this last step of the argument follow?

    I would be grateful for any help!

    Peter
    Last edited by Bernhard; September 6th 2011 at 04:53 AM.
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