Torsion group

In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which each element has finite order. All finite groups are periodic (cf. cyclic). The exponent of such a group, if extant, is the least common multiple of the orders of elements. The exponent exists for any finite group, and it divides the group order.

Significance

Burnside's problem is a classical question, which deals with the relationship between periodic groups and finite groups, if only finitely-generated group groups are considered: 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 groups. Another example is the direct sum of all dihedral groups. 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, see Golod–Shafarevich theorem, and by Aleshin and Grigorchuk using automata.

Mathematical logic

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

${\displaystyle \forall x.\,((x=e)\lor (x\circ x=e)\lor ((x\circ x)\circ x=e)\lor \cdots )}$

which contains an infinite disjunction 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 implies that no set of first-order formulae can characterize the periodic groups.[1]

Related notions

The torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. A torsion abelian group is an abelian group in which every element has finite order. A torsion-free abelian group is an abelian group in which the identity element is the only element with finite order.