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Math Help - group homomorphism

  1. #1
    Member Mauritzvdworm's Avatar
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    group homomorphism

    Let G\text{ and }H be additive subgroups of \mathbb{R} both containing 1. Let \eta:G\rightarrow H be an order preserving group homomorphism such that \eta(1)=1. Show that \eta(g)=g for all g\in G.

    The idea that I am working with is to let s,t\in G such that
    s-t>1 then there exists an integer m such that
    s>m>t.

    now apply \eta and get the following
    \eta(s)-\eta(t)>1 and \eta(s)>\eta(m)>\eta(t)
    and suppose that \eta(g)>g for all g\in G. In this way I hope to get some contradiction

    Or is there another way?
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  2. #2
    Member Mauritzvdworm's Avatar
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    This problem is still bugging me... does anybody have any ideas?
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