# show that G is abelian....

• Mar 17th 2014, 01:45 AM
Aman3230
show that G is abelian....
If G is a group such that (ab)= a2b2 forevery a,b Є G, show that G is abelian.
• Mar 17th 2014, 03:56 AM
FernandoRevilla
Re: show that G is abelian....
Quote:

Originally Posted by Aman3230
If G is a group such that (ab)= a2b2 forevery a,b Є G, show that G is abelian.

Surely you meant $(ab)^2=a^2b^2.$ Using that the group product is cancellable:

$$(ab)^2=a^2b^2\Rightarrow abab=aabb\Rightarrow bab=abb\Rightarrow ab=ba\;(\forall a,b\in G.)$$
• Mar 17th 2014, 04:20 AM
Aman3230
Re: show that G is abelian....

in my text book only (ab)= a2b2 was given anyway

thanks alot 4 giving ur precious time
• Mar 17th 2014, 04:35 AM
Deveno
Re: show that G is abelian....
If ab = a2b2 for every a,b in G, then it certainly is the case for a = e, the identity of G.

Then we have:

eb = e2b2

b = b2

e = b, for EVERY b in G, which means that G is a group with only one element: e.
• Mar 17th 2014, 04:41 AM
Aman3230
Re: show that G is abelian....
thanks