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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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    @DonAntonio I just saw that too... I just registered with an email so I guess I neglected that. Going through old questions now, thanks for the note.2012-11-19
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    Anyway: sometimes it is necessary to "do some dirty work", so I'd advice you to explicitly evaluate the commutator of two elements in each case and try to find out some pattern. In some cases you'll get a pretty easy result (e.g., $\,\mathcal O_2\,$ is abelian , so its commutator subgroup is...), in others some apparent pattern appears (triangular upper matrices, say).2012-11-19
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    @DonAntonio Thanks. I will follow your advice. This business just seems like walking in the dark.2012-11-19
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    @Benjamin: and why is that a bad thing? If you already knew what you were going to find then you wouldn't be learning anything.2012-11-19
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    @QiaochuYuan Oh no I agree. What I meant was I'm not necessarily a big fan of trying to squeeze out patterns by experimenting, even though in many cases it is precisely the thing to do.2012-11-19
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    @Benjamin: well, if you don't currently have another way to start...2012-11-19
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    A useful fact: the commutator subgroup is the smallest subgroup $H$ of $G$ such that $G/H$ is abelian.2012-11-19
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    By the way, if you just answered "the set of all commutators", then it would not necessarily be correct. The commutator subgroup is the subgroup generated by all commutators, and sometimes contains elements that are not themselves commutators.2012-11-19
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    @DerekHolt Oh I was talking about the generating set, as you can see.2012-11-20

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