- Periodic group
group theoryin 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 grouphas an exponent: it is a divisor of |"G"|. Burnside's problemis 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).
Infinite examples of 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. None of these examples has a finite generating set. Explicit examples of finitely generatedinfinite periodic groups were constructed by Golod, based on joint work with Shafarevich, and by Aleshin and Grigorchuk using automata.
* 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).
PlanetMatharticles on [http://planetmath.org/encyclopedia/PeriodicGroup.html periodic groups] and [http://planetmath.org/encyclopedia/Exponent.html exponent] .
Wikimedia Foundation. 2010.
Look at other dictionaries:
Group theory — is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The… … Wikipedia
Group 6 element — Group → 6 ↓ Period 4 24 Cr 5 … Wikipedia
Group 3 element — || Group 3/ungrouped | 7 || **ActinidesThe Group 3 elements are chemical elements comprising the third vertical column of the periodic table. IUPAC has not recommended a specific format for the periodic table, so different conventions are… … Wikipedia
Group 7 element — Group → 7 ↓ Period 4 25 Mn 5 … Wikipedia
Group 9 element — Group → 4 ↓ Period 4 27 Co 5 … Wikipedia
Group 10 element — Group 10 redirects here. For the rugby league competition, see Group 10 Rugby League. Group → 4 ↓ Period 4 … Wikipedia
Group 11 element — Group → 11 ↓ Period 4 29 Cu 5 … Wikipedia
Group — can refer to: Sociology * Group action (sociology) * Group behaviour * Groups of people, a description of various different human groups ** Peer group ** Workgroup * Group dynamics * Group (sociology), a sub set of a culture or of a society *… … Wikipedia
group — [gro͞op] n. [Fr groupe < It gruppo, a knot, lump, group < Gmc * kruppa, round mass: see CROP] 1. a number of persons or things gathered closely together and forming a recognizable unit; cluster; aggregation; band [a group of houses] 2. a… … English World dictionary
Group 8 element — Group 8 redirects here. For the Swedish organization, see Group 8 (Sweden). A Group 8 element is one in the series of elements in group 8 (IUPAC style) in the periodic table, which consists of the transition metals iron (Fe), ruthenium (Ru),… … Wikipedia