Periodic group

This article is about the mathematical concept. For groups in the chemical periodic table, see Group (periodic table).

In group theory, 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, although all finite cyclic 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).

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 union 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, and by Aleshin and Grigorchuk using automata.

Mathematical logic

One of the interesting properties of periodic groups is that they cannot be formalized in terms of first-order logic. This is because doing so would require an axiom of the form

\forall x.\, ((x=e) \lor (x\circ x=e) \lor ((x\circ x)\circ x=e) \lor \ldots)

which contains an infinite disjunction and is therefore inadmissible. It is 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 torsion groups.[1]

See also

References

  1. ↑ Ebbinghaus, H.-D.; Flum, J.; Thomas, W. (1994). Mathematical logic (2. ed., 4. pr. ed.). New York [u.a.]: Springer. p. 50. ISBN 978-0-387-94258-2. Retrieved 18 July 2012. However, in first-order logic we may not form infinitely long disjunctions. Indeed, we shall later show that there is no set of first-order formulas whose models are precisely the torsion groups.

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