Torsion group

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 that divides 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 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 that 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.