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Concept
Torsion group
Summary
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple 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.
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 groups. Another example is the direct sum of all dihedral groups. 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, and by Aleshin and Grigorchuk using automata. These groups have infinite exponent; examples with finite exponent are given for instance by Tarski monster groups constructed by Olshanskii.
Burnside's problem
Burnside's problem is a classical question which deals with the relationship between periodic groups and finite groups, when only finitely-generated groups 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 groups, the answer to Burnside's problem restricted to the class is positive.
One of the interesting properties of periodic groups is that the definition cannot be formalized in terms of first-order logic. This is because doing so would require an axiom of the form
which contains an infinite disjunction and is therefore inadmissible: First order logic permits quantifiers over one type and cannot capture properties or subsets of that type.
Official source
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Related concepts (13)
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 element with finite order. While finitely generated abelian groups are completely classified, not much is known about infinitely generated abelian groups, even in the torsion-free countable case. Abelian group An abelian group is said to be torsion-free if no element other than the identity is of finite order.
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 torsion module is a module that equals its torsion submodule. A module is torsion-free if its torsion submodule comprises only the zero element. This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements.
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 periodic group) if every element of A has finite order and is called torsion-free if every element of A except the identity is of infinite order. The proof that AT is closed under the group operation relies on the commutativity of the operation (see examples section).