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Here's the full question:

Determine the commutator subgroups of the following groups:

a) $SO_2$

b) $O_2$

c) the group $M$ of isometries of the plane

d) $S_n$

e) $SO_3$

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.

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    @DerekHolt Oh I was talking about the generating set, as you can see.2012-11-20

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Just a few hints on some of the subproblems:

c) Note that the commutator of two isometries is always orientation preserving because the commutator always consists of an even number of orientation-inverting factors. Thus $[M,M]$ can contain only rotations and translations. In fact $[M,M]$ contains at least all rotations and all translations: The rotation by half the angle or the translation by half the distance can be written as product of two reflections; the comuutator of these reflections is then the given arbitrary rotation or translation (because the reflections are their own inverse).

d) Note that $[a,b]$ is always an even permutation, i.e. $[S_n,S_n]\le A_n$. Do you know a set of generators for $A_n$? COuld you maybe write these as commutators?