Finite exponent of compact torsion groups (bounded exponent of periodic compact groups)

Does every periodic compact Hausdorff group have bounded exponent — that is, for every such compact group, one in which each element has finite order, must there exist a single positive integer $N$ with $x^N=1$ for all elements $x$? The question goes back to Herfort (1979) and is the open exponent-companion to Zelmanov's 1992 theorem, built on Wilson's 1983 structure theorem and the classification of finite simple groups, that every periodic compact group is locally finite. By Herfort's reduction it is equivalent to asking whether every torsion pro-p group has finite exponent. All known classes of examples do have finite exponent, yet neither a proof nor a counterexample is known; as Wilson noted in 2023, the problem is essentially about countably based pro-p groups that are not finitely generated, where the restricted Burnside theorem yields only local finiteness.

Background

A compact group is a topological group whose underlying space is compact and Hausdorff; the basic examples are closed subgroups and arbitrary products of finite groups, and by the Peter–Weyl theorem (the fundamental representation-theoretic structure theorem for compact groups) every compact group is an inverse limit of compact Lie groups. A group is periodic (equivalently, torsion) if every element has finite order, and it has bounded exponent (equivalently, finite exponent) if the orders of its elements are bounded overall: there is a positive integer $N$ with $x^N=1$ for every element $x$. A group is locally finite if every finitely generated subgroup is finite. Bounded exponent does not imply finiteness, and periodic does not imply bounded exponent.

For compact groups these notions interact rigidly. A nontrivial compact connected group contains elements of infinite order, so every periodic compact group is totally disconnected, and a compact totally disconnected group is a profinite group: an inverse limit of finite groups, taken with the inverse-limit (product) topology. Thus 'periodic compact group' and 'torsion profinite group' denote the same objects. A pro-p group, for a prime $p$, is an inverse limit of finite p-groups (groups in which every element has order a power of $p$); profinite groups admit Sylow subgroups analogous to those of finite groups, one pro-p Sylow for each prime.

The problem belongs to the lineage of the Burnside problems. The general Burnside problem asked whether a finitely generated torsion group must be finite; the answer is no — Golod and Shafarevich produced finitely generated infinite residually finite p-groups, and Grigorchuk and Gupta–Sidki produced such groups that are moreover torsion. (A group is residually finite if the intersection of its subgroups of finite index is trivial, so that it embeds in its profinite completion.) The restricted Burnside problem asked whether, for fixed numbers of generators and a fixed exponent bound, there are only finitely many finite groups; Zelmanov answered it affirmatively in 1990–92. A consequence is that every profinite group of bounded exponent is locally finite.

Without an exponent bound, the compact questions are harder. V. P. Platonov asked whether every compact torsion group is locally finite. Wilson (1983) proved the structure theorem: a compact torsion group has a finite series of closed characteristic subgroups in which each factor is either a pro-p group or a Cartesian product of isomorphic finite simple groups. Zelmanov (1992) proved that a torsion pro-p group is locally finite — using the Lie-algebra techniques of his restricted Burnside solution — and combined this with Wilson's theorem and the classification of finite simple groups to conclude that every periodic compact group is locally finite, settling Platonov's problem; he also proved that every infinite compact group contains an infinite abelian subgroup.

Local finiteness, however, does not bound exponents: a locally finite torsion group is a directed union of finite subgroups whose exponents may grow without bound, and the question is whether this can occur inside a compact group. The question goes back at least to Herfort (1979). Two reductions shape it. First, Herfort proved that the order of a compact torsion group is divisible by only finitely many primes (1979), and, more generally, that a profinite group has nontrivial p-Sylow subgroups for only finitely many primes if and only if this is true for its procyclic subgroups (1982) — a procyclic group being one topologically generated by a single element. This yields that a torsion profinite group has finite exponent if and only if its Sylow subgroups do, so the problem reduces to the pro-p case: every periodic compact group has bounded exponent if and only if every torsion pro-p group has finite exponent. Second, Wilson (2023) observes that the problem is essentially about countably based pro-p groups that are not finitely generated — countably based meaning that the topology has a countable base of open neighbourhoods of the identity — since a finitely generated pro-p group that is torsion is already finite, by the restricted Burnside theorem.

The standard constructions do not settle the question. A Cartesian product of finite p-groups is compact, and it is torsion exactly when the exponents of the factors are bounded: if element orders in the factors grow unboundedly, the product contains an element of infinite order. The profinite completions of the known finitely generated residually finite torsion p-groups of Grigorchuk and Gupta–Sidki fail as well, since their pro-p completions contain elements of infinite order, so the completions are not torsion. Positive results are known for special classes — soluble (solvable) periodic profinite groups, compact torsion groups with an open set of elements of the same prime order (Khukhro, 1989), and torsion pro-p groups admitting a fixed-point-free automorphism of finite order (Shumyatsky, 2000) all have finite exponent — but none of the methods extends to the general case.

Problem Statement

Let $G$ be a compact Hausdorff topological group that is periodic (equivalently, torsion): every element of $G$ has finite order. Must $G$ have bounded exponent — that is, must there exist a positive integer $N$, depending on $G$, such that $x^N=1$ for every $x$ in $G$?

Equivalently: must every torsion profinite group have finite exponent? By Herfort's reduction (Herfort, 1979, 1982 — the element orders of a torsion profinite group involve only finitely many primes, and the exponent of the group divides the product of the exponents of its Sylow subgroups), the question is further equivalent to: must every torsion pro-p group, for a prime $p$, have finite exponent?

The question is settled affirmatively by a proof that a uniform exponent bound exists for each periodic compact group, or negatively by an explicit construction of a single periodic compact group whose element orders are unbounded. Note that Zelmanov's theorem already gives that every such $G$ is locally finite, that is, a directed union of finite subgroups; the question is exactly whether the exponents of those finite subgroups can grow without bound inside a compact group.

References

1. E. I. Zelmanov, 'On periodic compact groups', Israel Journal of Mathematics 77 (1992), no. 1–2, 83–95. DOI: 10.1007/BF02808012. https://doi.org/10.1007/BF02808012
2. J. S. Wilson, 'On the structure of compact torsion groups', Monatshefte für Mathematik 96 (1983), no. 1, 57–66. DOI: 10.1007/BF01298934. https://doi.org/10.1007/BF01298934
3. W. N. Herfort, 'Compact torsion groups and finite exponent', Archiv der Mathematik 33 (1979), no. 1, 404–410. DOI: 10.1007/BF01222776. https://doi.org/10.1007/BF01222776
4. W. Herfort, 'An arithmetic property of profinite groups', Manuscripta Mathematica 37 (1982), no. 1, 11–17. DOI: 10.1007/BF01239941. https://doi.org/10.1007/BF01239941
5. E. I. Khukhro, 'A comment on periodic compact groups', Siberian Mathematical Journal 30 (1989), no. 3, 493–496. DOI: 10.1007/BF00971509. https://doi.org/10.1007/BF00971509
6. P. Shumyatsky, 'On pro-p groups admitting a fixed-point-free automorphism', Journal of Algebra 228 (2000), no. 1, 357–366. DOI: 10.1006/jabr.1999.8267. https://doi.org/10.1006/jabr.1999.8267
7. P. Shumyatsky, 'Profinite groups in which many elements have prime power order', Journal of Algebra 562 (2020), 188–199. DOI: 10.1016/j.jalgebra.2020.07.003. https://doi.org/10.1016/j.jalgebra.2020.07.003 (arXiv:2001.01263.)
8. J. S. Wilson, 'Profinite groups with few conjugacy classes of elements of infinite order', Archiv der Mathematik 120 (2023), no. 6, 557–563. DOI: 10.1007/s00013-023-01851-4. https://doi.org/10.1007/s00013-023-01851-4 (arXiv:2209.14753.)
9. E. Zelmanov, 'Lie algebras and torsion groups with identity', Journal of Combinatorial Algebra 1 (2017), no. 3, 289–340. DOI: 10.4171/JCA/1-3-2. https://doi.org/10.4171/JCA/1-3-2 (arXiv:1604.05678.)
10. C. Acciarri and P. Shumyatsky, 'On profinite groups in which commutators are covered by finitely many subgroups', Mathematische Zeitschrift 274 (2013), no. 1–2, 239–248. DOI: 10.1007/s00209-012-1067-z. https://doi.org/10.1007/s00209-012-1067-z (arXiv:1112.5879.)
11. Y. Barnea, R. Camina, M. Ershov and M. L. Lewis, 'On groups that can be covered by conjugates of finitely many cyclic or procyclic subgroups', Mathematische Annalen 392 (2025), no. 2, 1747–1779. DOI: 10.1007/s00208-025-03119-4. https://doi.org/10.1007/s00208-025-03119-4 (arXiv:2210.15746.)
12. C. Acciarri and P. Shumyatsky, 'Profinite groups with restricted centralizers of powers', Annali di Matematica Pura ed Applicata (published online 2026). DOI: 10.1007/s10231-026-01704-1. https://doi.org/10.1007/s10231-026-01704-1 (arXiv:2602.18839.)
13. J. Fariña-Asategui and S. Radi, 'Torsion elements in branch pro-p groups', Journal of Algebra and Its Applications 25 (2026), no. 9, article 2650288; published online July 2025. DOI: 10.1142/S0219498826502889. https://doi.org/10.1142/S0219498826502889 (arXiv:2407.01746.)
14. E. I. Zel'manov, 'Solution of the restricted Burnside problem for groups of odd exponent', Mathematics of the USSR-Izvestiya 36 (1991), no. 1, 41–60. DOI: 10.1070/IM1991V036N01ABEH001946. https://doi.org/10.1070/IM1991V036N01ABEH001946
15. E. I. Zel'manov, 'A solution of the restricted Burnside problem for 2-groups', Mathematics of the USSR-Sbornik 72 (1992), no. 2, 543–565. DOI: 10.1070/SM1992V072N02ABEH001272. https://doi.org/10.1070/SM1992V072N02ABEH001272
16. R. I. Grigorchuk, 'Bernside's problem on periodic groups' (the published English title carries this typo; the work is usually cited as 'On Burnside's problem on periodic groups'), Functional Analysis and Its Applications 14 (1980), no. 1, 41–43. DOI: 10.1007/BF01078416. https://doi.org/10.1007/BF01078416
17. N. Gupta and S. Sidki, 'On the Burnside problem for periodic groups', Mathematische Zeitschrift 182 (1983), no. 3, 385–388. DOI: 10.1007/BF01179757. https://doi.org/10.1007/BF01179757
18. E. S. Golod and I. R. Shafarevich, 'On the class field tower', Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 28 (1964), 261–272 (Russian); English translation: 'On class field towers', American Mathematical Society Translations, Series 2, 48 (1965), 91–102. DOI: 10.1090/trans2/048/05. https://doi.org/10.1090/trans2/048/05