# Prove G is abelian iff...

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• October 8th 2009, 07:38 PM
elninio
Prove G is abelian iff...
Let G be a group. Prove that G is abelian if and only if (ab)^-1 = a^-1 * b^-1 for all a,b in G.
• October 8th 2009, 09:49 PM
Gamma
Quote:

Originally Posted by elninio
Let G be a group. Prove that G is abelian if and only if (ab)^-1 = a^-1 * b^-1 for all a,b in G.

First note we always have $(ab)^{-1}=b^{-1}a^{-1}$ See why?

so if G is abelian, the forward direction is clear (just move them past eachother). Conversely, if we have $a^{-1}b^{-1}=(ab)^{-1}=b^{-1}a^{-1}$ for all a and b in G.

Just multiply on the left by (ba) and on the right by (ab)