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Periodic group

 

 
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Periodic group
 
 
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In group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
 in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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 Binary operation that combines any two of its element to form a third element....
 in which each element
Element (mathematics)

In mathematics, an element or member of a Set is any one of the distinct objects that make up that set....
 has finite
Finite set

In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
 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....
. 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 or monogenous group is a group that can be generating set of a group 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 ....
, 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 or lowest common multiple or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b....
, if it exists, of the orders of the elements of G. Any finite group
Finite group

In mathematics, a finite group is a group that has finite setly many elements. During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth: in particular, the local analysis of finite groups, and the theory of solvable groups and nilpotent groups....
 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 generating set of a group 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, a finite group is a group that has finite setly many elements. During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth: in particular, the local analysis of finite groups, and the theory of solvable groups and nilpotent groups....
s, if we assume only that G is a finitely-generated group.






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In group theory
Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as group .The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring , field , and vector spaces can all be seen as groups endowed with additional operations and axioms....
 in mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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 Binary operation that combines any two of its element to form a third element....
 in which each element
Element (mathematics)

In mathematics, an element or member of a Set is any one of the distinct objects that make up that set....
 has finite
Finite set

In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
 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....
. 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 or monogenous group is a group that can be generating set of a group 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 ....
, 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 or lowest common multiple or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b....
, if it exists, of the orders of the elements of G. Any finite group
Finite group

In mathematics, a finite group is a group that has finite setly many elements. During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth: in particular, the local analysis of finite groups, and the theory of solvable groups and nilpotent groups....
 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 generating set of a group 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, a finite group is a group that has finite setly many elements. During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth: in particular, the local analysis of finite groups, and the theory of solvable groups and nilpotent groups....
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).

Infinite examples of 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 group theory, the Pr?fer p-group or the p-quasicyclic group or p8-group, Z, for a prime number p is the unique torsion subgroup in which every element has p pth roots....
s. None of these examples has a finite generating set. Explicit examples of finitely generated
Finitely generated

Finitely generated may refer to:* finitely generated group* finitely generated abelian group* finitely generated module* finitely generated algebra...
 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 problems which they are able to solve. Automata theory is closely related to formal language theory as the automata are often classified by the class of formal languages they are able to recognize....
.

External links

  • PlanetMath
    PlanetMath

    PlanetMath is a free content, collaborative, online mathematics encyclopedia. The emphasis is on peer review, rigour, openness, pedagogy, real-time content, interlinked content, and community....
     articles on and .


See also


  • torsion (algebra)
  • 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 ....