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Subgroup Question

  1. Jan 25, 2012 #1
    If ab=ba in a group G, let H={g[itex]\in[/itex]G|agb=bga}.

    Show that H is a subgroup.


    I have the identity because ab=ba implies that aeb=bea[itex]\in[/itex]H.

    I cant seem to figure out how to get the inverses or closer. Any suggestions or slight nudges in the right direction?
     
  2. jcsd
  3. Jan 25, 2012 #2

    vela

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    Let x, y ∈ H. You want to show that xy-1 ∈ H, so consider a(xy-1)b.

    a(xy-1)b = ax(bb-1)y-1(a-1a)b = (axb)(b-1y-1a-1)(ab)

    Use what you know about a, b, x, and y to show that that product is equal to b(xy-1)a.
     
  4. Jan 25, 2012 #3
    Thanks that was clever.... didnt think of adding A's and B's in the middle and inversing.
     
  5. Jan 25, 2012 #4

    vela

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    It's a pretty common technique to create combinations you know how to deal with. Good to have in your toolkit.
     
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