1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    pasmith

    User Avatar
    Homework Helper

    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

    pasmith

    User Avatar
    Homework Helper

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Subgroup inverse map question
  1. Inverse map (Replies: 0)

  2. Subgroup Question (Replies: 0)

  3. Subgroup Question (Replies: 10)

  4. Subgroup Question (Replies: 3)

Loading...