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?