Introduction to group theory/Right cancellation

Proof

Let $G$ be a group and let $a,b,c \in G$ such that $ac=bc$ . Since $c \in G, \exist c^{-1} \in G$ such that $c*c^{-1}=e$ . Multiplying $ac=bc$ on each side by $c^{-1}$ we obtain. $ac*c^{-1}=bc*c^{-1}$ . Applying the definition of inverses ( $c*c^{-1}=e$ ) we get $a*e=b*e$ Applying the definition of identity we get $a=b$

Q.E.D.

This article is issued from Wikiversity - version of the Wednesday, July 17, 2013. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.