## Presentations of finite groups [closed]

If $G$ is a group of order $n$, with presentation $\langle X_1, X_2, \cdots, X_r \colon R_1, R_2,\cdots, R_l\rangle$, and if we add more relation in the presentation, is there a possibility of getting a different group of same order.

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No -- the resulting group is a quotient of $G$, so it is either smaller than $G$ or isomorphic to $G$. – Andy Putman Feb 22 2011 at 6:28