Periodic group

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In group theory in mathematics, a periodic group is a group in which each element has finite order. All finite groups are periodic.

The exponent of a periodic group G is the least common multiple, if it exists, of the orders of the elements of G. Any finite group has an exponent: it is a divisor of |G|.

Burnside's problem is a classical question, which deals with the relationship between periodic groups and finite groups, 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).

A group constructed by Grigorchuk is an interesting example of an infinite periodic group.

For the multiplicative group of integers modulo n, the exponent is given by the Carmichael function.


  • 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).

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