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Math Help - order preserving group homomorphism

  1. #1
    Member Mauritzvdworm's Avatar
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    order preserving group homomorphism

    say you have two subgroups G\text{ and }H and an order preserving group homomorphism \eta:G\rightarrow H which maps the identity of G to the identity of H

    Can we show that \eta(g)=g for all g\in G so that we can identify G as a subgroup of H and \eta is then just the inclusion?
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    Quote Originally Posted by Mauritzvdworm View Post
    say you have two subgroups G\text{ and }H and an order preserving group homomorphism \eta:G\rightarrow H which maps the identity of G to the identity of H

    Can we show that \eta(g)=g for all g\in G so that we can identify G as a subgroup of H and \eta is then just the inclusion?

    The condition \eta(g)=g\,,\,\forall g\in G is very strong, and it'd mean G\subset H (i.e. not only as identification but as actual subset of a set...).

    But from the given data yes: \eta(g)=1\Longleftrightarrow g=1\Longleftrightarrow \eta\,\,is\,\,1-1 and thus G is embedded in H.

    Tonio
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