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Periodic group
From Wikipedia, the free encyclopedia
In group theory in mathematics, a periodic group or a torsion group is a group in which each element has finite order. All finite groups are periodic. The concept of a periodic group should not be confused with that of a cyclic group.
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).
Explicit examples of finitely generated infinite periodic groups were constructed by Golod, based on joint work with Shafarevich, and by Aleshin and Grigorchuk using automata.
For the multiplicative group of integers modulo n, the exponent is given by the Carmichael function.
- E. S. Golod, On nil-algebras and finitely approximable p-groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 273--276.
- N. V. Aleshin, Finite automata and the Burnside problem for periodic groups, (Russian) Mat. Zametki 11 (1972), 319--328.
- R. I. Grigorchuk, On Burnside's problem on periodic groups, Functional Anal. Appl. 14 (1980), no. 1, 41--43.
- 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).
- PlanetMath articles on periodic groups and exponent.
