Let phi: G ---> H be a group homomorphism. Show that phi[G] is abelian if and only if for all x, y in G, we have xyx^(-1)y^(-1) in ker(phi).
Proving (=>) seems almost obvious since if it is abelian that means xyx^(-1)y^(-1) = xx^(-1)yy^(-1)=ee which is in the kernel. Please show how to do the reverse (<=) and show that phi is abelian.© BrainMass Inc. brainmass.com November 30, 2021, 1:28 am ad1c9bdddf
"=>": If phi(G) is abelian, then ...
Group homomorphism and abelian groups are investigated in the solution.