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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in the field of
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, a locally finite group is a type of group that can be studied in ways analogous to a
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
. 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 In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
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: * Every finite group is locally finite * Every infinite
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
of finite groups is locally finite (Although the direct product may not be.) * The Prüfer groups are locally finite abelian groups * Every Hamiltonian group is locally finite * Every periodic solvable group is locally finite . * Every
subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation  ...
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 In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
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. * Omega-categorical groups (that is, groups whose first-order theory characterises them up to isomorphism) are locally finite Non-examples: * No group with an element of infinite order is a locally finite group * No nontrivial
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...
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 . 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 .


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