An infinite group described by finitely many rules, with only finite-order elements

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Family 247 · Group theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

A group element has finite order if repeating it sufficiently many times returns to the identity. If every element has this property, can the group nevertheless be infinite?

The answer has long been yes, even when finitely many elements generate the group. The harder question asks whether finitely many defining relations can also suffice. A finite list of rules must then force every possible combination of generators to have finite order, while leaving infinitely many distinct combinations alive.

Family 247 claims such a construction. Its earlier manuscript builds an infinite finitely presented periodic group and an accompanying algebra with unusual nilpotence properties. A later companion strengthens the group result to an infinite finitely presented residually finite $2$-group: every element has power-of-two order, and finite quotients can distinguish every nonidentity element.

The orders are not uniformly bounded. That qualification is indispensable, because existing theorems rule out the stronger bounded-exponent conclusion in this residually finite setting.

2. Background and essential definitions

Generators, relations and finite-order behavior

A group is finitely generated if a finite list of elements and their inverses produces every element by multiplication. A finite presentation

$$G=\langle s_1,\ldots,s_k\mid r_1,\ldots,r_m\rangle$$

adds a finite list of words required to equal the identity. All other equalities must follow from those relations and the group axioms.

For example, $\langle s\mid s^N=1\rangle$ is a finite cyclic group. With several generators, imposing finite order on each generator does not automatically impose finite order on their products. That is why an infinite periodic group is much harder to describe than a collection of generators that happen to be torsion elements.

Periodic means that for every $g\in G$, some positive integer $n(g)$ satisfies $g^{n(g)}=1$. Bounded exponent means one integer works for every element. A $2$-group has element orders $1,2,4,8,\ldots$, possibly with no common upper bound.

The presentations here are ordinary presentations of abstract discrete groups. They do not conceal an additional infinite family of periodicity axioms, and they are not presentations in the category of topological pro-$2$ groups.

Finite quotients can see an infinite group

A group is residually finite when each nonidentity element has a nonidentity image in some finite group. Different elements may require different finite quotients. Thus residual finiteness provides an unlimited collection of finite windows onto an infinite object.

If every element of the original group has power-of-two order, the same holds in every finite image. Such a finite image is a finite $2$-group. There is no implication that a single finite image captures the entire original group.

Nil is weaker than nilpotent

An associative algebra is a vector space with a bilinear associative multiplication. It need not be commutative, and it need not have an identity element.

A nonunital algebra $I$ is nil if every element $a$ satisfies $a^{N(a)}=0$ for some exponent depending on $a$. It is nilpotent if there is one $N$ for which every product of $N$ elements vanishes. Nilpotence is therefore a uniform statement about mixed products, not just repeated copies of one element.

Adjoining an identity gives the unitization $R=\mathbb F_2\oplus I$, where $\mathbb F_2$ is the two-element field. If $a$ is nilpotent, then $1+a$ is invertible: a finite geometric series provides its inverse. This is the elementary bridge between nil algebras and groups.

The construction needs more than elementwise nilpotence. It claims that every finite square matrix with entries in $I$ is nilpotent, with an exponent allowed to depend on the matrix. The paper calls this property matrix-nil.

3. Development history and previous partial results

Burnside’s 1902 questions initiated the study of how finite generation interacts with finite element orders. Several distinct problems emerged, and their different conclusions should not be conflated.

Golod’s 1964 work, using the Golod–Shafarevich dimension argument, produced infinite finitely generated periodic groups. It also gave residually finite groups with torsion restricted to one prime. The underlying nil-algebra method already connected algebraic nilpotence to group-theoretic finite orders.

Novikov and Adian constructed infinite finitely generated groups of bounded exponent. Their result answered a stronger torsion question in one direction, but did not supply the additional ordinary finite presentation sought here.

Grigorchuk gave a concrete infinite finitely generated residually finite $2$-group. Its original construction is not finitely presented. It therefore demonstrates that residual finiteness and power-of-two torsion can coexist, while leaving the finite-presentation requirement open.

Zel’manov’s solution of the restricted Burnside problem goes in a different direction. In particular, a finitely generated residually finite group with a fixed power-of-two exponent is finite. Any example with the new manuscript’s other properties must consequently have elements of arbitrarily large order.

Ol’shanskii and Sapir constructed finitely presented torsion-by-cyclic groups. Those groups have infinite cyclic quotients, so they are not periodic. Their work shows how powerful finite-presentation constructions can be without resolving the fully periodic case.

There is a parallel algebraic history. Golod’s nil algebras already refuted the broad finite-generation form of Kurosh’s algebraic finiteness question. The remaining strengthening asks for finitely many defining algebra relations. Ufnarovskij’s question about finitely presented nil algebras, and Amitsur’s related question for Jacobson radical algebras, impose this additional constraint.

The new sources also credit finite local-rule constructions. Durand, Romashchenko and Shen developed hierarchical simulations by tile sets. Ivanov-Pogodaev and Kanel-Belov constructed a finitely presented infinite nilsemigroup. A semigroup result controls individual words; the present algebra construction additionally needs cancellation for arbitrary sums and matrices. That additional step is essential to the claimed group construction.

4. What the new manuscripts claim

M2: the algebra and the periodic group

The manuscript dated 23 September 2026 constructs an infinite-dimensional, finitely presented, graded algebra

$$R=\mathbb F_2\oplus I,\qquad I=\bigoplus_{d\geq1}R_d,$$

with each homogeneous component $R_d$ finite-dimensional and with $I$ matrix-nil. Grading means multiplication adds degrees.

It asserts that $I$, viewed as a nonunital algebra, is finitely presented, nil and not nilpotent. It is also Jacobson radical: every $1-a$ is invertible in the unitization. The unital algebra $R$ is finitely presented, algebraic over $\mathbb F_2$, and infinite-dimensional. Algebraic means each element satisfies some nonzero polynomial over the field; the polynomial degree may depend on the element.

From this ring it constructs

$$\Gamma=\operatorname{St}_{12}(R),$$

an infinite ordinarily finitely presented periodic group. The notation denotes a Steinberg group, an abstract group encoding elementary matrix operations and their relations. Its relation to actual matrices has a kernel that must be controlled separately.

M2 also applies an existing theorem of Ershov and Jaikin-Zapirain to give $\Gamma$ Kazhdan’s property $(T)$, a rigidity condition for almost invariant vectors in unitary representations. Since $\Gamma$ is infinite, this implies nonamenability.

M1: residual finiteness and a fixed torsion prime

The companion dated 5 October 2026 uses M2’s algebra and hierarchy as essential inputs. It asserts that $\Gamma$ is residually finite and considers

$$G=\ker\bigl(\operatorname{St}_{12}(R)\longrightarrow \operatorname{St}_{12}(\mathbb F_2)\bigr).$$

The target is finite, so $G$ has finite index in $\Gamma$. The claimed result is that $G$ is infinite, ordinarily finitely presented and residually finite, with every element of finite power-of-two order.

The subgroup step matters: M2’s whole periodic group and M1’s $2$-group are not identical statements. M1 strengthens the construction, rather than providing an unrelated alternative proof of M2’s algebra theorem.

5. Why the advance matters

Importance: 5/5, provisional. The claimed construction resolves the finitely presented Burnside question and several parallel finite-presentation problems for algebras. It combines properties that earlier examples achieved only in weaker combinations.

A finite presentation is a particularly strong form of finite description. The proposed example would show that finitely many rules can force periodicity of every group element without forcing the whole group to be finite.

Residual finiteness adds another significant constraint: the group’s infinitude is compatible with separating all elements by finite images. The result therefore cannot be understood merely as a group whose unusual behavior becomes invisible in every finite quotient.

On the algebra side, the same construction distinguishes elementwise nilpotence from uniform nilpotence even under finite presentation. Its matrix-nil property supplies a bridge powerful enough to support the group theorem.

6. The main idea

Finite rules generate a hierarchy

The algebra uses a finite alphabet, word replacements that preserve length, and declarations that certain words equal zero. Nonzero replacement classes form a vector-space basis. A surviving word of large degree witnesses that the algebra has not collapsed to finite dimension.

The rules encode larger-scale letters by blocks of smaller-scale letters. Clean encodings preserve nonzero words through the hierarchy, producing survivors at arbitrarily large degrees. Yet the construction must also control every allowed word, including malformed intermediate verification states.

For sufficiently long nonzero words, the source obtains a decomposition into bounded end pieces and recognizable tokens. Long runs of specially colored letters can be permuted. In characteristic two, summing suitable matrix weights over these permutations produces cancellation.

A weighted automaton keeps track of how tokens can be segmented when passing between hierarchy levels. The paper uses increasing palette sizes to make the cancellation dominate the increased matrix dimension. A separate bookkeeping variable distinguishes the number of multiplied factors from their word lengths. The resulting claim is nilpotence of each fixed matrix over $I$, not merely of each basis word.

Matrices give torsion, but their abstract lifts need more

For a nilpotent matrix $A$ in characteristic two,

$$(1+A)^{2^k}=1+A^{2^k}=1$$

once $2^k$ exceeds its nilpotence index. This explains the source of power-of-two torsion near the identity.

The elementary matrix group over $R$ is infinite because its off-diagonal entries can vary through the infinite ring. Existing finite-presentation results for Steinberg groups turn the finite ring presentation into a finite group presentation.

However, a torsion matrix image does not ensure that its abstract preimage is torsion. M2 therefore controls the central kernel of the map from the Steinberg group to the elementary matrix group. It compares substitutions of cyclic and truncated shift matrices into a graded expression. Rational homological stability then brings the resulting torsion information back to the fixed rank twelve.

Finite truncations must also detect the kernel

Discarding sufficiently high degrees gives finite rings $R/R_{\geq d}$. These truncations readily detect a nonzero matrix entry. The harder issue is an element of the abstract group that maps to the identity matrix.

M1 assigns start and end phases to sufficiently long nonzero words, modulo arbitrarily large odd integers $L$. Compatible phases constrain products and allow roots to be moved between phase-labeled coordinates.

Suppose a central element escapes every degree truncation. One calculation makes a sufficiently long cyclic transfer vanish. The phase calculation identifies the same transfer with the $L$-th power of that element. It already has power-of-two order, while $L$ is odd, so these conclusions force it to be trivial.

A stabilization theorem connects this stable calculation back to rank twelve. Finite truncations then separate the whole group, and the finite-index kernel of reduction to $\mathbb F_2$ gives the final $2$-group.

7. Scope, limitations, and open questions

The construction uses characteristic two and produces a $2$-group. It does not state an analogous ordinary finite-presentation theorem for every prime.

The element orders are unbounded, and the nilpotence indices of algebra elements and matrices can grow. Replacing any of these element-dependent bounds by one global bound would change the problem substantially.

The matrices are over a special noncommutative ring. This is not a finite-dimensional linear representation over a field that contradicts classical finiteness theorems for finitely generated periodic linear groups.

Finite presentation is an existence conclusion with a construction by hierarchical rules. The sources do not turn it into a small practical multiplication table for the infinite group. Likewise, residual finiteness asserts the existence of detecting finite quotients, without the digest supplying efficient bounds on their size.

Finally, the algebra and group conclusions have distinct logical requirements. Surviving words give infinitude; matrix cancellation gives periodicity of the matrix image; central-kernel arguments supply torsion and finite-quotient detection for the abstract group. Omitting the last stage would leave the strongest claims unsupported by the stated construction.

8. Sources and further reading

Burnside, On an unsettled question in the theory of discontinuous groups (1902), is the original source. Golod’s nil-algebra and finite-approximation paper (1964) and Grigorchuk’s On Burnside’s problem on periodic groups (1980) give earlier infinite residually finite torsion examples.

Zel’manov’s restricted Burnside theorem for 2-groups explains why the new group cannot have bounded exponent. Ol’shanskii and Sapir’s torsion-by-cyclic construction addresses a different finite-presentation problem.

For the construction’s inputs, see Durand, Romashchenko and Shen, Fixed-point tile sets and their applications; Krstić and McCool, Presenting GLn(k⟨T⟩); and Weibel, Module structures on the K-theory of graded rings.

Source manuscripts

• M1. OpenAI, An infinite finitely presented residually finite 2-group (October 5, 2026).
• M2. OpenAI, An infinite finitely presented periodic group (September 23, 2026).

Sources consulted on 22 October 2026 at repository revision 1581d4c. Download the manuscript citations.