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!

Homework Help: Prove (a^(-1))^(-1) = a

  1. Dec 17, 2012 #1


    User Avatar
    Homework Helper

    1. The problem statement, all variables and given/known data

    Trying to run through every problem i can in my book in preparation for my exam. I've solved this one before, but it slipped my mind how to do it :


    2. Relevant equations

    Working in a group, so group axioms I suppose.

    3. The attempt at a solution

    I forgot where to start this one off, I was thinking :

    e = aa-1
    a-1e = a-1aa-1

    That wont get me anywhere though, any pointers would be appreciated.
  2. jcsd
  3. Dec 17, 2012 #2
    Do you know that every element in a group has a unique inverse?? You can use this by showing that [itex]a[/itex] and [itex](a^{-1})^{-1}[/itex] are inverses of the same element. So they must be equal.
  4. Dec 17, 2012 #3


    User Avatar
    Homework Helper

    Ah, so what you're saying is if a is a group element, then it has an inverse which is also a group element denoted by a-1.

    Since a-1 is also a group element and the inverse of a, then (a-1)-1 is also a group element and is the inverse of a-1.
  5. Dec 17, 2012 #4
    Yes. So [itex]a^{-1}[/itex] has two inverses. Those inverses must equal.
  6. Dec 17, 2012 #5

    I like Serena

    User Avatar
    Homework Helper

    Hi Zondrina!

    In a group every element has to have an inverse.
    Now suppose we have b=a-1.
    Then according to the group axioms we have: ab=e
    What happens if you multiply the left and right hand sides with b-1?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook