Here's the full question:
Determine the commutator subgroups of the following groups:
c) the group $M$ of isometries of the plane
f) the group $G$ of $3 \times 3$ upper triangle matrices with 1's along the diagonal over the prime field $F_p$
I have little to no intuition for commutator subgroups. I imagine I need to find the generating set for each, but I suppose I can't just say "the set of all commutators". Then they want something explicit. But then where do you start? I was thinking I could start with the commutators of generators of each group. Would that be the right track, and where should I go from there if it is?
Any help much appreciated.