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Math Help - Subsets of an Image

  1. #1
    Newbie bakerconspiracy's Avatar
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    Subsets of an Image

    Hey,

    I'm having a bit of trouble with this problem:

    Let phi: (G,*)->(H, . ) be a homomorphism.

    If S is a subset of im(Phi), prove that the group S generates is a subgroup of im(Phi)

    I don't know where to go with this....

    Thanks in advance
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bakerconspiracy View Post
    Hey,

    I'm having a bit of trouble with this problem:

    Let phi: (G,*)->(H, . ) be a homomorphism.

    If S is a subset of im(Phi), prove that the group S generates is a subgroup of im(Phi)

    I don't know where to go with this....

    Thanks in advance
    This is worded weird. Maybe you mean given K\leqslant G we have that \phi(K)\leqslant \phi(G)?

    If that is the case:

    Clearly e\in K\implies \phi(e)\in \phi(K)

    Also, if a,b\in\phi(K) then a=\phi(c),b=\phi(d) for some c,d\in K. But, since K is a group we have that cd\in K\implies \phi(cd)=\phi(c)\phi(d)=ab\in \phi(K).

    Lastly, if a\in \phi(K) then a=\phi(c) for some c\in K and so c^{-1}\in K since it is in a group. Thus, \phi(c^{-1})=\phi(c)^{-1}=a^{-1}\in\phi(K)
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  3. #3
    Newbie bakerconspiracy's Avatar
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    I really appreciate the help, but that wasn't quite what I was asking.

    What I was asking is, if S is a subset (not subgroup) of G, then prove that <S> is a subgroup of G.

    <S> meaning the group generated by S.

    Thanks again for your help
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bakerconspiracy View Post
    I really appreciate the help, but that wasn't quite what I was asking.

    What I was asking is, if S is a subset (not subgroup) of G, then prove that <S> is a subgroup of G.

    <S> meaning the group generated by S.

    Thanks again for your help
    But now I'm confused because you're talking about S\subseteq G when earlier S\subseteq\phi(G)?

    Also, do you consider \left\langle S\right\rangle to the intersection of all groups containing S?
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  5. #5
    Newbie bakerconspiracy's Avatar
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    "Also, do you consider to the intersection of all groups containing ?"

    yes, the intersection of all subgroups containing S.

    "But now I'm confused because you're talking about when earlier ? "

    Is there a difference though? I mean what if I defined L = the image of Phi(G). Then S is a subset of L, prove that the group generated by the set S is a subgroup of L. Same question just reduced right?
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  6. #6
    Newbie bakerconspiracy's Avatar
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    Well thank you anyways
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