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If we have a presentation $\langle x_i: R_j\rangle$ of $G$,

(1) what does the presentation (in terms of the given generators and the given relations) of a subgroup of $G$ looks like? In particular, what does the presentation of a normal subgroup $N$ looks like?

(2) What does the presentation of the quotient group $G/N$ looks like?

It seems that a subgroup should have less generators and the same relations, a quotient group should have same generators but more relations. Is this true?

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  • $\begingroup$ Yes you can always find a presentation of a quotient group with the same generators and more relations. But a subgroup is more complicated, and in general it might need more generators, not less. There is a general method called the Reidemeister-Schreier algorithm for finding a presentation of a subgroup. $\endgroup$ – Derek Holt Oct 31 '16 at 19:27
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(1) It depends. Removing some generators, and keeping all relations will give you a subgroup (you have to translate all relations involving the removed generators, you cannot simply remove them).

EDIT: Actually, "translating" the relations to a smaller set of generators won't work in general at all. Thanks to Derek.

Also, you can not create all subgroups in this way. A subgroup might in fact need more generators than the whole group. For example in the case without any relations (i.e. the free group), there is a nice (and possibly counter-intuitive) theorem that $\langle x_1,...,x_n:\emptyset\rangle$ is a subgroup of $\langle y_1, y_2 :\emptyset\rangle$ for any $k$.

(2) yes. Just add all relations of $N$ and you get the factor group $G/N$.

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    $\begingroup$ In your answer to (1), you may need to add extra relations to get a presentation of the subgroup generated by a subset of the generators. It is even possible for the original $R$ to be finite, but for the subgroup not to be finitely presented. $\endgroup$ – Derek Holt Oct 31 '16 at 20:20
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If you want more detail for (2), there's a really good proof (which gives an idea of how to construct such a presentation) of presentations of factor groups on pages 42/43 of D.L.Johnson's book on presentations

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