The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.
Examples: Non-examples: The class of locally finite groups is closed under subgroups, quotients, and extensions (Robinson 1996, p. 429).
Locally finite groups satisfy a weaker form of Sylow's theorems.
If there are finitely many conjugates, then the number of conjugates is congruent to 1 modulo p. In fact, if every countable subgroup of a locally finite group has only countably many maximal p-subgroups, then every maximal p-subgroup of the group is conjugate (Robinson 1996, p. 429).
While this need not be true in general, a result of Philip Hall and others is that every infinite locally finite group contains an infinite abelian group.