In group theory

Group theory

In mathematics and 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 as groups endowed with additional operations and...

, a periodic group or a torsion group is a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 in which each element has finite order

Order (group theory)

In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

. All finite groups are periodic. The concept of a periodic group should not be confused with that of a cyclic group

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

, although all finite cyclic groups are periodic.

The exponent of a periodic group G is the least common multiple

Least common multiple

In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...

, if it exists, of the orders of the elements of G. Any finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

 has an exponent: it is a divisor of |G|.

Burnside's problem

Burnside's problem

The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group...

 is a classical question, which deals with the relationship between periodic groups and finite group

Finite group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...

s, 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).

Examples of infinite periodic groups include the additive group of the ring of polynomials over a finite field, and the quotient group of the rationals by the integers, as well as their direct summands, the Prüfer group

Prüfer group

In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z, for a prime number p is the unique p-group in which every element has p pth roots. The group is named after Heinz Prüfer...

s. Another example is the union of all dihedral group

Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...

s. None of these examples has a finite generating set, and any periodic linear group with a finite generating set is finite. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich, and by Aleshin and Grigorchuk using automata

Automata theory

In theoretical computer science, automata theory is the study of abstract machines and the computational problems that can be solved using these machines. These abstract machines are called automata...

.

See also

• Torsion
• Torsion subgroup
  Torsion subgroup
  In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order...
  
  
• Jordan–Schur theorem
  Jordan–Schur theorem
  In mathematics, the Jordan–Schur theorem also known as Jordan's theorem on finite linear groups is a theorem in its original form due to Camille Jordan...
  
  
• Golod–Shafarevich theorem
  Golod–Shafarevich theorem
  In mathematics, the Golod–Shafarevich theorem was proved in 1964 by two Russian mathematicians, Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which has consequences in various branches of algebra.-The inequality:...
  
  

External links

• PlanetMath
  PlanetMath
  PlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be comprehensive, the project is hosted by the Digital...
  
  articles on periodic groups and exponent.