In group theory in mathematics, a periodic group is a group in which each element has finite order. All finite groups are periodic. Group theory is that branch of mathematics concerned with the study of groups. ... Euclid, detail from The School of Athens by Raphael. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an element (also called a member) is an object contained in a set (or more generally a class). ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

The exponent of a periodic group G is the least common multiple, if it exists, of the orders of the elements of G. Any finite group has an exponent: it is a divisor of |G|. In arithmetic and number theory the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. ... In mathematics, a finite group is a group which has finitely many elements. ...

Burnside's problem is a classical question, which deals with the relationship between periodic groups and finite groups, if we assume only that G is a finitely-generated group. The question is whether specifying an exponent forces finiteness (to which the answer is 'no', in general). One of the oldest open problems in group theory was first posed by William Burnside in a paper published in 1902. ... In mathematics, a finite group is a group which has finitely many elements. ...

A group constructed by Grigorchuk is an interesting example of an infinite periodic group.

For the multiplicative group of integers modulo n, the exponent is given by the Carmichael function. In mathematics, the multiplicative group of integers modulo n is the group defined by multiplication of the units (that is, the numbers relatively prime to ) in the ring for a given integer . ... In number theory, the Carmichael function of a positive integer , denoted , is defined as the smallest integer such that for every integer that is coprime to . ...

References

• R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means., Izv. Akad. Nauk SSSR Ser. Mat. 48:5 (1984), 939-985 (Russian).

External link

• PlanetMath article on periodic groups