Does every element of the derived series of a prop group is also a prop group?
The problem reduces to showing that every element of the derived series is a closed subgroup...But is it always true?
Hope you'll be able to help me
Thanks !
Does every element of the derived series of a prop group is also a prop group? The problem reduces to showing that every element of the derived series is a closed subgroup...But is it always true? Hope you'll be able to help me Thanks ! 


It seems this can't be expected. From Simons' thesis (p. xii):
I haven't Roman'kov's paper at hand yet and don't know about details of his example. The reference is: Roman'kov: The width of verbal subgroups of solvable groups. Algebra i Logika 21(1),6072(1982). In contrast, the groups in the lower central series of a finitely generated profinite group are always closed. This is proved by NikolovSegal in this paper (Theorem 1.4). 

