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?
Best Answer
(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 $G/N$.