Let $C$ and $D$ be non-abelian groups. Show that $C\oplus D$ doesn't satisfy the universal property of direct sum.
I think I must assume that the universal property is true and then use the free product to define a homomorphism $f:C\oplus D\to C*D $, and then get a contradiction, but I don't know how to do this, or how to use that the groups $C$ and $D$ are not abelian.
Any help will be very appreciated! Thank you so much!
The assumption that $C$ and $D$ are nonabelian is a red herring; all that is needed is that they are nontrivial. Indeed, let $c\in C$ and $d\in D$ be non-identity elements, and let $f:C\oplus D\to C*D$ be the canonical homomorphism that would be given by the universal property. Then in $C*D$, $$cd=f(c,1)f(1,d)=f(c,d)=f(1,d)f(c,1)=dc.$$ This is a contradiction because $cd$ and $dc$ are distinct reduced words and thus are distinct elements of $C*D$ (here we use the assumption that $c$ and $d$ are not the identity to conclude that $cd$ and $dc$ are reduced).