Results 1 to 2 of 2

Math Help - Abelian Group Proof

  1. #1
    Junior Member
    Joined
    Aug 2008
    Posts
    33

    Abelian Group Proof

    Prove that a group G is Abelian if and only if (ab)^-1 = (a^-1)(b^-1) for all a and b in G.

    I know how to do the first part of the proof.
    If G is Abelian then (ab)^-1 = (a^-1)(b^-1).
    (ab)^-1 = (b^-1)(a^-1)=(a^-1)(b^-1), since G is abelian.

    The second part is where I am snagged.
    If (ab)^-1 = (a^-1)(b^-1) then G is abelian.
    (ab)^-1 = (b^-1)(a^-1) = (ab)^-1 = (a^-1)(b^-1) Therefore (b^-1)(a^-1)=(a^-1)(b^-1). So G must be abelian.

    I'm not sure if i can make these statements. It seems like i'm assuming stuff.
    Any help is appreciated.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,792
    Thanks
    1687
    Awards
    1
    Given that \left( {ab} \right)^{ - 1}  = a^{ - 1} b^{ - 1} the following is true:
    \begin{gathered}<br />
  \left( {ab} \right)^{ - 1} b = a^{ - 1}  \hfill \\<br />
  \left( {ab} \right)^{ - 1} ba = e \hfill \\<br />
  ba = ab \hfill \\ <br />
\end{gathered}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Permutations and Abelian Group proof.
    Posted in the Advanced Algebra Forum
    Replies: 18
    Last Post: February 18th 2010, 07:43 AM
  2. Is the subgroup of an abelian group always abelian?
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: December 6th 2009, 11:38 PM
  3. abelian group
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: May 26th 2009, 11:59 AM
  4. Group Theory: Proof on abelian and isomorphic groups.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 20th 2009, 06:46 PM
  5. Abelian Group proof
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 9th 2009, 09:17 PM

Search Tags


/mathhelpforum @mathhelpforum