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Homework Help: Subgroup inverse map question

  1. Feb 13, 2014 #1
    1. The problem statement, all variables and given/known data

    For a group [itex]G[/itex] consider the map [itex]i:G\rightarrow G , i(g)=g^{-1}[/itex]
    For a subgroup [itex]H\subset G[/itex] show that [itex]i(gH)=Hg^{-1}[/itex] and [itex]i(Hg)=g^{-1}H[/itex]
    2. Relevant equations

    3. The attempt at a solution

    I know that for [itex] g_1,g_2 \in G[/itex] we have [itex]i(g_1g_2)=(g_1g_2)^{-1}=g_2^{-1}g_1^{-1}[/itex]
    Then since for any [itex]h\in H, h\in G [/itex] we have [itex]i(g_1h)=(g_1h)^{-1}=h^{-1}g_1^{-1}[/itex]
    Is this a good approach to the problem?
  2. jcsd
  3. Feb 13, 2014 #2


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    Working out what [itex]i(gh)[/itex] is for [itex]h \in H[/itex] is certainly a good start.
  4. Feb 13, 2014 #3
    Sorry I should have said I'm actually stuck at this point. Any pointers or hints would be appreciated :)
  5. Feb 13, 2014 #4


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    You are asked to show that, if [itex]H[/itex] is a subgroup of [itex]G[/itex], then for all [itex]g \in G[/itex], [itex]i(gH) = Hg^{-1}[/itex].

    So far you have that if [itex]h \in H[/itex] and [itex]g \in G[/itex] then [itex]i(gh) = h^{-1}g^{-1}[/itex]. You now need to explain why [itex]h^{-1}g^{-1} \in Hg^{-1}[/itex].
  6. Feb 13, 2014 #5
    since [itex]H[/itex] is a subgroup, any [itex]h\in H [/itex] has an inverse element [itex]h^{-1}\in H [/itex] such that [itex]hh^{-1}=h^{-1}h=e[/itex] hence [itex]h^{-1}g^{-1}\in Hg^{-1}[/itex]
    Last edited: Feb 13, 2014
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