mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, in the field of

group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...

, a locally finite group is a type of

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

that can be studied in ways analogous to a

finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or ma ...

Sylow subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...

Carter subgroup In mathematics, especially in the field of group theory, a Carter subgroup of a finite group ''G'' is a self-normalizing subgroup of ''G'' that is nilpotent. These subgroups were introduced by Roger Carter, and marked the beginning of the post ...

s, 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, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marke ...

. Since the

cyclic subgroup In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses. In oth ...

s of a locally finite group are finitely generated hence finite, every element has finite

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...

, 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. To see how the direct sum is used in abstract algebra, consider a mor ...

of finite groups is locally finite (Although the

direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...

may not be.) * Omega-categorical groups * The

Prüfer group In mathematics, specifically in group theory, the Prüfer ''p''-group or the ''p''-quasicyclic group or ''p''∞-group, Z(''p''∞), for a prime number ''p'' is the unique ''p''-group in which every element has ''p'' different ''p''-th roots. ...

s are locally finite abelian groups * Every

Hamiltonian group In group theory, a Dedekind group is a group ''G'' such that every subgroup of ''G'' is normal. All abelian groups are Dedekind groups. A non-abelian Dedekind group is called a Hamiltonian group. The most familiar (and smallest) example of a Hamil ...

is locally finite * Every periodic solvable group is locally finite . * Every

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 invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

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 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''−1' ...

is locally finite * A

Tarski monster group In the area of modern algebra known as group theory, a Tarski monster group, named for Alfred Tarski, is an infinite group ''G'', such that every proper subgroup ''H'' of ''G'', other than the identity subgroup, is a cyclic group of order a fixe ...

is periodic, but not locally finite.

Properties

The class of locally finite groups is closed under subgroups, quotients, and

extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ex ...

. 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 Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups. Biography He was educated first at Christ's Hospital, where he won the Thomps ...

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 In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by . History conjectured that every nonabelian finite simple group has even order. suggested using t ...

on the solubility of finite groups of odd order .

References

* *

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