User m.z. - MathOverflow

Answer by M.Z. for Names of finite groups

2010-12-08T00:17:23Z

There is a useful convention to decorate some of the groups with an index which is the smallest $n$ for which the group can act transitively on $n$ points, i.e. embeds in $S_n$ as a transitive subgroup. The notation for $S_n, A_n, D_n, C_n$, your $Q_8$ and for example Mathieu groups $M_{11}, M_{12}, M_{22}$ (although not other sporadic simple groups) follow this pattern.

Of course, there is also another convention to use the size of the group instead...

Answer by M.Z. for Proving interesting theorems about S_n using its character table.

2010-12-01T22:12:58Z

Because all the entries in the character table are integers and not just algebraic integers, you get that a proof that every permutation $\sigma$ of order $n$ is conjugate to all $\sigma^j$ for $j$ coprime to $n$. (Of course, one usually uses this in the opposite direction, to deduce that all entries are integers!)

Answer by M.Z. for Best examples of physics providing insight into math

2010-11-29T20:44:34Z

I think one of the most intriguing examples is mirror symmetry: physical intuition has suggested formulae for the number of curves of a fixed genus ona Calabi-Yau 3-fold that mathematicians did not expect