Given a rank-2 group $G= < a,b> $ . 
Is it true and trivial that $ [G,G] = < [a,b], [b,a] > $ ? 


Thanks ! 



Commutator Subgroup - Group Theory
No, the derived subgroup of the free group of rank 2 is infinitely generated (as every normal subgroup of infinite index), see W. Magnus, A. Karras, D. Solitar, Combinatorial group theory.


