Work out all the details of the Euler-Rodrigues formula, by using the fact that any rotation in [math]\mathbb R^3[/math] has a rotation axis.

Work out the theory of the subgroups of [math]O_N,U_N[/math] constructed via

with [math]d\in\mathbb N\cup\{\infty\}[/math], which generalize both [math]O_N,U_N[/math] and [math]SO_N,SU_N[/math].

Look up the literature, and find the relevance of the symplectic groups, and of symplectic geometry in general, to questions in classical mechanics.

Find and then write down a brief account of the Shephard-Todd theorem, stating that the irreducible complex reflection groups are

along with a number of exceptional examples, more precisely [math]34[/math] of them.