  We study the uniform Roe algebras associated to locally finite groups. We
show that for two countable locally finite groups $\Gamma$ and $\Lambda$, the
associated uniform Roe algebras $C^*_u(\Gamma)$ and $C^*_u(\Lambda)$ are
$*$-isomorphic if and only if their $K_0$ groups are isomorphic as ordered
abelian groups with units. This can be seen as a non-separable non-simple
analogue of the Glimm-Elliott classification of UHF algebras. To the best of
our knowledge, this is the first classification result for a class of
non-separable unital $C^*$-algebras. Along the way we also obtain a rigidity
result: two countable locally finite groups are bijectively coarsely equivalent
if and only if the associated uniform Roe algebras are $*$-isomorphic.
  Finally, we give a summary of $C^*$-algebraic characterizations for (not
necessarily countable) locally finite discrete groups in terms of their uniform
Roe algebras. In particular, we show that a discrete group $\Gamma$ is locally
finite if and only if the associated uniform Roe algebra
$\ell^\infty(\Gamma)\rtimes_r \Gamma$ is locally finite-dimensional.
