Can a Lie group be discrete, or even finite?

Any (countable) set $X$ admits a unique smooth $0$-manifold structure: just give $X$ the discrete topology, and take the unique map $\{x\}\to\mathbb{R}^0$ as a chart for each $x\in X$, since $\mathbb{R}^0$ is just a single point. Compatibility of the charts is trivial since no two distinct charts overlap. Moreover, any map from a $0$-dimensional smooth manifold to any smooth manifold is automatically smooth (in local charts, you just have a map $\mathbb{R}^0\to\mathbb{R}^n$ which is always smooth, since there are no partial derivatives that need to exist). In particular, any group structure on $X$ makes $X$ a Lie group.

(The countability requirement is if you require manifolds to be second-countable, which is sometimes not done but generally is in the context of Lie groups.)