Locally Finite Group - Examples and Non-examples

Examples and Non-examples

Examples:

• Every finite group is locally finite
• Every infinite direct sum of finite groups is locally finite (Robinson 1996, p. 443) (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 (Dixon 1994, Prop. 1.1.5).
• Every subgroup of a locally finite group is locally finite. If G is a group and S is a subgroup of G and F is a finite subset of S, the subgroup generated by F cannot be an infinite subset of S for then it would be an infinite subset of G contradicting the fact that G is locally finite.
• Every group has a unique maximal normal locally finite subgroup (Robinson 1996, p. 436)
• 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.