  In this work, we prove that every SFT, sofic shift, and strongly irreducible
shift on locally finite groups has strong dynamical properties. These
properties include that every sofic shift is an SFT, every SFT is strongly
irreducible, every strongly irreducible shift is an SFT, every SFT is entropy
minimal, and every SFT has a unique measure of maximal entropy, among others.
In addition, we show that if every SFT on a group is strongly irreducible, or
if every sofic shift is an SFT, then the group must be locally finite, and this
extends to all of the properties we explore. These results are collected in two
main theorems which characterizes the local finiteness of groups by purely
dynamical properties. In pursuit of these results, we present a formal
construction of \textit{free extension} shifts on a group $G$, which takes a
shift on a subgroup $H$ of $G$, and naturally extends it to a shift on all of
$G$.
