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.

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 finite, every element has finite order, and so the group is periodic.

Examples and non-examples

Examples:
* Every finite group is locally finite
* Every infinite direct sum of finite groups is locally finite harv|Robinson|1996|p=443
* The Prüfer groups are locally finite abelian groups
* Every periodic solvable group is locally finite harv|Dixon|1994|loc=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 and infinite subset of "G" contradicting the fact that "G" is locally finite.
* If G is not a locally finite group, then it is possible that there is a locally finite subgroup of G; in particular, the trivial group.

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.

Properties

The class of locally finite groups is closed under subgroups, quotients, and extensions harv|Robinson|1996|p=429.

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 harv|Robinson|1996|p=429.

The class of locally finite groups behaves somewhat similarly to the class of finite groups. Much of the 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 harv|Dixon|1994|p=v..

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 harv|Robinson|1996|p=432.

References

External links

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