1 · The essentials
Can a mathematical system have infinitely many elements even when it has a finite list of building blocks and defining rules, and repeatedly combining any one element with itself eventually returns to the starting point? These manuscripts give a negative answer to the suggestion that such a system must be finite. In the main construction, the number of repetitions needed for each element is a power of two. There is no single upper bound on those numbers. Moreover, each element other than the starting point can be distinguished from it in some finite version of the system. A companion construction gives an infinite-dimensional algebra described by finitely many generators and relations, even though repeatedly multiplying any one of its elements eventually gives zero. The main result has a Lean formalization; this verification statement should not be extended automatically to every companion claim.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, think of an endless collection of repeating routines governed by a finite instruction sheet: each routine returns to its start after a power-of-two number of steps, but their lengths have no common bound. Finite inspection devices can distinguish each nontrivial routine from doing nothing, although no single device is asserted to capture the whole collection.
Why it matters
The construction separates finite description from finite size: even requiring every element to have finite order does not force an ordinarily finitely presented group to be finite. The main manuscript adds residual finiteness and restricts all element orders in its subgroup to powers of two. The companion manuscript supplies analogous negative answers to the stated finiteness and nilpotence questions for finitely presented algebras.
2 · Going further
The group-theoretic question is whether an infinite periodic group can have an ordinary finite presentation: finitely many generators and defining relations, with every element nevertheless having finite order. The companion manuscript constructs such a group, answering the finitely presented Burnside question negatively. The main manuscript studies this group's more specific form, , and establishes residual finiteness: every nonidentity element remains nonidentity under some homomorphism to a finite group.
The central object in the main conclusion is the finite-index subgroup . It is infinite, ordinarily finitely presented, and residually finite, and each of its elements has order for some nonnegative integer . These orders are unbounded. The argument described in the abstract uses phases of long words in the companion's graded algebra so that finite degree truncations detect every element of , including elements of its central kernel. The supplied abstracts do not explain the phase construction further.
On the algebra side, the companion manuscript shows the existence of an infinite-dimensional, finitely presented, nonunital nil associative -algebra that is Jacobson radical but not nilpotent. Here each individual element has a power equal to zero, but there is no common product length at which all products vanish. Its unitization, obtained by adjoining an identity, is finitely presented, algebraic, and infinite-dimensional. These are the companion's negative answers to the finitely presented nil- and radical-algebra nilpotence questions and to the finite-presentation version of Kurosh's algebraic finiteness question.
Useful notions
- Groups and the order of an element
- Generators and defining relations
- Homomorphisms, kernels, and finite quotients
- Associative algebras and vector-space dimension
- Nil elements versus nilpotent algebras
What was known before / what is new
Before
The source frames the finitely presented Burnside question as asking whether finite presentation together with finite order for every element forces a group to be finite. It also identifies corresponding nilpotence questions for finitely presented nil and radical algebras, and a finite-presentation version of Kurosh's algebraic finiteness question. No further historical background or earlier partial results are supplied.
New
The companion manuscript shows the existence of an infinite ordinarily finitely presented periodic group and the stated infinite-dimensional finitely presented algebras. The main manuscript establishes residual finiteness of the companion's Steinberg group. Its finite-index subgroup gives the stronger group conclusion: an infinite ordinarily finitely presented residually finite group whose elements all have finite, but unbounded, power-of-two orders.
Limits and assumptions
The power-of-two order conclusion is stated for G, while the larger group Γ is described as periodic and residually finite. Finite order is an element-by-element condition here, not a uniform bound on orders; similarly, the companion's nil algebra is explicitly not nilpotent. The algebra constructions are over the two-element field, and the nil algebra initially has no identity. The main result has a Lean formalization, but the supplied description does not specify its detailed scope or establish that every companion algebra claim is formalized. The abstracts give only a brief account of the truncation argument and identify no further remaining open questions.
Mini glossary
- Finite presentation —
- A description of a group or algebra using finitely many generators and finitely many defining relations.
- Periodic group —
- A group in which every element has finite order.
- Residual finiteness —
- The property that every nonidentity group element has a nonidentity image in some finite group.
- Nil algebra —
- An algebra in which every element has some positive power equal to zero.
- Nilpotent algebra —
- An algebra for which there is a fixed positive length such that every product of that many elements is zero.
- Algebraic algebra —
- An algebra in which each element satisfies a nonzero polynomial equation over the underlying field.
Keywords : Finite presentation · Residual finiteness · Nil algebras
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽2-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
An infinite finitely presented residually finite 2-group
05/10/2026We prove that the infinite, ordinarily finitely presented periodic Steinberg group of a companion paper is residually finite. Its finite-index subgroup is infinite, ordinarily finitely presented, and residually finite, and every element of G has finite 2-power order. The orders of its elements are unbounded. Phases of long words in the companion's graded algebra allow finite degree truncations to detect all elements of Γ, including the central kernel.
An infinite finitely presented periodic group
23/09/2026We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively. We also construct an infinite-dimensional finitely presented nonunital nil associative algebra over 𝔽2 that is Jacobson radical but not nilpotent. Its unitization is finitely presented, algebraic, and infinite-dimensional. These algebras answer the finitely presented nil- and radical-algebra nilpotence questions and the finite-presentation version of Kurosh's algebraic finiteness question negatively.