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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, in the field of
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
, a locally finite group is a type of
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
that can be studied in ways analogous to a
finite group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
.
Sylow subgroup In mathematics, specifically in the field of finite group theory In abstract algebra, a finite group is a group (mathematics), group whose underlying set is finite set, finite. Finite groups often arise when considering symmetry of mathema ...
s, 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 group, finitely generated subgroup is finite group, finite. Since the cyclic subgroups of a locally finite group are finitely generated hence finite, every element has finite order (group theory), order, and so the group is periodic group, periodic.


Examples and non-examples

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 group, 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.


Properties

The class of locally finite groups is closed under subgroups, quotient group, quotients, and group extension, extensions . Locally finite groups satisfy a weaker form of Sylow's theorems. If a locally finite group has a finite p-group, ''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 group, 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 .


References

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External links

*{{springer, id=L/l060410, author=A.L. Shmel'kin *Otto H. Kegel and Bertram A. F. Wehrfritz (1973),
Locally Finite Groups
', Elsevier Properties of groups Infinite group theory