Group structures on Hausdorff space

Question

Could anyone give me some practical (and possibly intuitive) examples of Group structures on Hausdorff spaces? Let us say you had to get freshmen university students interested into fields of maths they hardly ever heard of, and you would like to call their attention by drawing some everyday life examples (or sort of) which they could be fascinated by. Thanks in advance.

Answer 1

Here's a list of standard examples for you. I've opted to make the examples as general as I can so that the list can be more mercifully short.

• $S^1=\{z\in\Bbb C : |z|=1\}$ group structure: multiplication of complex numbers
• $F^*= F\setminus \{0\}$, $F$ a field. group structure: multiplication of field elements
• $V$ any normed vector space. Group structure: addition of vectors
• $\mathrm{GL}_n(V)$ and $\mathrm{SL}_n(V)$, a locally compact vector space. Group structure: multiplication
• $G$ arbitrary with the discrete topology.
• Profinite groups with all groups in the system being given the discrete topology.
• $G/H$ where $G$ is any Hausdorff group and $H$ is a closed subgroup.

Note that the vector space example includes fields and the matrix rings over fields, similarly the arbitrary group with the discrete topology includes $\Bbb Z$, and subgroups are always winners, since subspaces of Hausdorff spaces are Hausdorff. In particular $\Bbb R^+$ follows from the $F^*$ example. I included the $S^1$ despite it being a subgroup of $\Bbb C^*$ because it's so central to the theory. Other notables include the Möbius tranforms on $\Bbb C$, which are related to projective special linear groups, themselves quotients of general linear groups by the closed subgroup of diagonal matrices.