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Math Help - residually finite problem

  1. #1
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    residually finite problem

    i need to show that subgroups of residually finite groups are residually finite.
    i have 2 definitions for residually finite.

    definition 1: A group G is called residually finite if for all nontrivial element in G, there exists a normal subgroup with finite index N_x in G such that x \notin N_x.

    definition 2: A group G is called residually finite if for all nontrivial element x in G, there is a homomorphism f from G to a finite group such that f(x)\ne1.

    NonCommAlg proved the above by using definition 2.
    now i try to prove by using definition 1. im stuck somewhere and quite confused where to continue. please give me some hints.
    here is my proof:

    Let G be a residually finite group.
    Let H be a subgroup of G and 1 \ne x \in H.
    By def. 1, there exists a normal subgroup with finite index N in G such that x \notin N.
    Let N' be a normal subgroup with finite index in H.
    Since intersection of subgroups is also a subgroup, i have N' as a subgroup of G.
    So, now i have N \vartriangleleft G and N'<G.
    I apply 2nd isomorphism theorem here and obtained N \cap N' ={\{1\}} \triangleleft N' and N'/(N \cap N') \cong NN'/N.

    I don't know whether I followed the correct path, but i'm stuck here.
    I want to prove that x \notin N'. Please give me some hints.
    Thank you.
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  2. #2
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    Quote Originally Posted by deniselim17 View Post

    Let G be a residually finite group. Let H be a subgroup of G and 1 \ne x \in H. By def. 1, there exists a normal subgroup with finite index N in G such that x \notin N.
    so N'=N \cap H is a normal subgroup of H and x \notin N'. the only thing we need to prove now is that [H:N']< \infty: define f: \frac{H}{N'} \longrightarrow \frac{G}{N} by f(hN')=hN. obviously f is a one-to-one group

    homomorphism. thus \frac{H}{N'} is isomorphic to a subgroup of \frac{G}{N}, which is a finite group because N is finite index. hence \frac{H}{N'} has to be finite too, i.e. [H:N']< \infty.
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