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. 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: Every finite group is locally finite Every infinite direct sum of finite groups is locally finite (Although the direct product may not be.) Omega-categorical groups The Prüfer groups are locally finite abelian groups Every Hamiltonian group is locally finite Every periodic solvable group is locally finite . Every subgroup of a locally finite group is locally finite. (Proof. Let G be a locally finite group and S a subgroup. Every finitely generated subgroup of S is a (finitely generated) subgroup of G.) Hall's universal group is a countable locally finite group containing each countable locally finite group as subgroup. Every group has a unique maximal normal locally finite subgroup Every periodic subgroup of the general linear group over the complex numbers is locally finite. Since all locally finite groups are periodic, this means that for linear groups and periodic groups the conditions are identical. Non-examples: No group with an element of infinite order is a locally finite group No nontrivial free group is locally finite A Tarski monster group is periodic, but not locally finite. 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.