Locally finite group

In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studied. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.
Definition and first consequences.
A locally finite group is a group for which every finitely generated subgroup is finite.
Since the cyclic subgroups of a locally finite group are finitely generated hence finite, every element has finite order, and so the group is periodic.
Examples and non-examples.
Examples:
Non-examples:
Properties.
The class of locally finite groups is closed under subgroups, quotients, and extensions .
Locally finite groups satisfy a weaker form of Sylow's theorems. If a locally finite group has a finite "p"-subgroup contained in no other "p"-subgroups, then all maximal "p"-subgroups are finite and conjugate. 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 .
The class of locally finite groups behaves somewhat similarly to the class of finite groups. Much of the 1960s theory of formations and Fitting classes, as well as the older 19th century and 1930s theory of Sylow subgroups has an analogue in the theory of locally finite groups .
Similarly to the Burnside problem, mathematicians have wondered whether every infinite group contains an infinite abelian subgroup. 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. The proof of this fact in infinite group theory relies upon the Feit–Thompson theorem on the solubility of finite groups of odd order .