1. Limited time only! Sign up for a free 30min personal 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!

Prove that (a^-1)^-1=a

  1. Nov 3, 2013 #1
    1. The problem statement, all variables and given/known data

    Let (G,[itex]\circ[/itex]) be a group. Show that [itex]\forall[/itex]a[itex]\in[/itex]G [tex](a^{-1})^{-1} = a[/tex]
    3. The attempt at a solution

    I came up with the following. [itex]a^{-1}[/itex] is the inverse of [itex](a^{-1})^{-1}[/itex], therefore we have:

    [tex]a^{-1} \circ (a^{-1})^{-1} = e[/tex]
    But for [itex]a^{-1} \circ (a^{-1})^{-1}[/itex] to be equal to e, it has to be the case that:
    [tex]a^{-1} \circ a = e[/tex]
    , therefore [itex](a^{-1})^{-1}[/itex] has to be equal to a.

    Now I think the proof is incorrect, but I'm not sure. I think the mistake is trying to imply that [itex](a^{-1})^{-1}[/itex] is equal to a, just because both [itex]a^{-1} \circ (a^{-1})^{-1}[/itex] and [itex]a^{-1} \circ a[/itex] are equal to a. Is this a valid proof technique?
  2. jcsd
  3. Nov 3, 2013 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Well, isn't a group associative?

    We have:

    so that:
    a(ring)(a^-1(ring)(a^-1)^-1))=a(ring)e (*)

    Invoking associativity on LHS in (*) should do the trick.
  4. Nov 3, 2013 #3
    Ah yes, didn't think of that option. Thanks a lot for your help.

    In general if you try to prove equalities like this, what should you think about? Should you just consider all definitions that are valid and then just try some things out until you come up with the desired equality or what should the process of a finding a proof for a statement like this look like?
  5. Nov 3, 2013 #4


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    There isn't any foolproof method to kill all the problems you might encounter.

    But, as you do more of such problems, your brain sort of figures out the structure, and clever ideas might start popping up by themselves.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted