Periodic Group
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In
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 branch of
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 ...
, a torsion group or a periodic group is a
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 ...
in which every element has
finite order In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is ''infinite''. The ''order'' of an element of a group (also called period length or period) is the order of the sub ...
. The exponent of such a group, if it exists, is the
least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bot ...
of the orders of the elements. For example, it follows from Lagrange's theorem that every finite group is periodic and it has an exponent dividing its order.


Infinite examples

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 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. Another example is the direct sum of all
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...
s. None of these examples has a finite generating set. Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich, see
Golod–Shafarevich theorem In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can b ...
, and by Aleshin and Grigorchuk using
automata An automaton (; plural: automata or automatons) is a relatively self-operating machine, or control mechanism designed to automatically follow a sequence of operations, or respond to predetermined instructions.Automaton – Definition and More ...
. These groups have infinite exponent; examples with finite exponent are given for instance by
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 fixed ...
s constructed by Olshanskii.


Burnside's problem

Burnside's problem is a classical question which deals with the relationship between periodic groups and
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 marked ...
s, when only
finitely-generated group In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of s ...
s are considered: Does specifying an exponent force finiteness? The existence of infinite, finitely generated periodic groups as in the previous paragraph shows that the answer is "no" for an arbitrary exponent. Though much more is known about which exponents can occur for infinite finitely generated groups there are still some for which the problem is open. For some classes of groups, for instance
linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a f ...
s, the answer to Burnside's problem restricted to the class is positive.


Mathematical logic

One of the interesting properties of periodic groups is that the definition cannot be formalized in terms of
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. This is because doing so would require an axiom of the form :\forall x,\big((x = e) \lor (x\circ x=e) \lor ((x\circ x)\circ x=e) \lor \cdots\big) which contains an infinite
disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...
and is therefore inadmissible: First order logic permits quantifiers over one type and can't capture properties or subsets of that type. It is also not possible to get around this infinite disjunction by using an infinite set of axioms: the
compactness theorem In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally no ...
implies that no set of first-order formulae can characterize the periodic groups.


Related notions

The
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 (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group (or ...
of an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
''A'' is the subgroup of ''A'' consisting of all elements that have finite order. A
torsion abelian group In abstract algebra, a torsion abelian group is an abelian group in which every element has finite order.Dummit, David; Foote, Richard. ''Abstract Algebra'', , pp. 369 For example, the torsion subgroup of an abelian group is a torsion abelian grou ...
is an abelian group in which every element has finite order. A
torsion-free abelian group In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only e ...
is an abelian group in which the identity element is the only element with finite order.


See also

*
Torsion (algebra) In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A to ...
*
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. In that form, it states that there is a function ''ƒ''(''n'') such that given a finite subgr ...


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). Properties of groups {{Abstract-algebra-stub