If G is a group such that (ab)= a2b2 forevery a,b Є G, show that G is abelian.
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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:
$$(ab)^2=a^2b^2\Rightarrow abab=aabb\Rightarrow bab=abb\Rightarrow ab=ba\;(\forall a,b\in G.)$$
thanks 4 reply.....
in my text book only (ab)= a2b2 was given anyway
thanks alot 4 giving ur precious time
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.
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