# Abelian Groups

• September 25th 2008, 05:00 PM
JCIR
Abelian Groups
Suppose that G is a group with the property that for every choice of elements in G, axb=cxd implies ab=cd. Prove that G is abelian.
"Middle cancelation" implies commutativity.
• September 25th 2008, 05:34 PM
NonCommAlg
Quote:

Originally Posted by JCIR
Suppose that G is a group with the property that for every choice of elements in G, axb=cxd implies ab=cd. Prove that G is abelian.
"Middle cancelation" implies commutativity.

$a(a^{-1}b^{-1})b = e(a^{-1}b^{-1})ba.$