# Math Help - Abelian Group Proof

1. ## 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.

2. Given that $\left( {ab} \right)^{ - 1} = a^{ - 1} b^{ - 1}$ the following is true:
$\begin{gathered}
\left( {ab} \right)^{ - 1} b = a^{ - 1} \hfill \\
\left( {ab} \right)^{ - 1} ba = e \hfill \\
ba = ab \hfill \\
\end{gathered}$