we-solve (beta)

Math-1147 · mathematics · Group theory

Which lattices occur as intervals in subgroup lattices of finite groups?

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

A finite lattice is a nonempty finite partially ordered set $L$ in which every pair $x,y$ has a greatest lower bound $x\wedge y$ and a least upper bound $x\vee y$. It has a least and a greatest element. An isomorphism of lattices is a bijection preserving the order in both directions, equivalently preserving these two operations.

For a finite group $G$ and a subgroup $H\le G$, the subgroup interval is the ordered set \[ [H,G]=\{K:H\le K\le G\}, \] ordered by inclusion. It is a lattice: the meet of $K_1,K_2$ is $K_1\cap K_2$, and their join is the subgroup $\langle K_1\cup K_2\rangle$ generated by their union. The bottom and top are $H$ and $G$. No normality assumption on $H$ or on the intermediate subgroups is made.

The conjecture recorded here is the negative representation assertion \[ \exists\text{ finite lattice }L\ \forall\text{ finite groups }G \ \forall H\le G:\qquad L\not\cong[H,G]. \] Thus the task is to exhibit a finite lattice and prove that no such interval represents it, or to refute the conjecture by proving that every finite lattice is representable. Restricting a search to groups of bounded order cannot establish nonrepresentability.

An interval $[H,K]$ inside the subgroup lattice of a larger finite group gives the same object by taking $K$ itself as ambient group, so allowing two proper interval endpoints does not enlarge the representation class. This is an interval problem, not the stronger requirement that $L$ be an entire subgroup lattice.

Short English statement

Is there a finite lattice that cannot be realized as the interval of all subgroups between H and G in any finite group?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1147-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1147

Theorem 1.2 supplies a finite lattice excluded from every full subgroup interval [H,G] of every finite group, with no normality or group-order restriction, exactly proving the record’s assertion. Theorem 1.1(iv) proves subgroup-interval representability undecidable, which rules out every lattice being representable because that would give the constant yes algorithm, hence proves the record’s negative existence assertion. Within these assumptions, this is a claimed affirmative answer to the stated question.

Which lattices occur as intervals in subgroup lattices of finite groups?

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: Some finite lattice is claimed to have no realization as the complete congruence lattice of any finite algebra; A colored-graph criterion is claimed to characterize realizable finite congruence lattices, with an undecidable recognition problem; recognition of finite subgroup intervals is also claimed to be undecidable. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
The two existence claims concern finite lattices. First, some nonempty finite lattice $L$ satisfies \[ \forall A\text{ finite nonempty algebra of finite signature},\qquad L\not\cong\operatorname{Con}(A). \] Second, there is a finite lattice $L'$ for which \[ \forall G\text{ finite},\ \forall D\le G,\qquad L'\not\cong[D,G]. \] Here $\operatorname{Con}(A)$ is the congruence lattice and $[D,G]$ is the complete interval of subgroups containing $D$.
Result 2.
For a nonempty finite lattice $L$, let $\mathcal R(L)$ mean that $L\cong\operatorname{Con}(A)$ for some finite algebra $A$. The claimed finite colored-graph criterion is equivalent to $\mathcal R(L)$, with a finite decision procedure once the candidate graph size is fixed. Nevertheless, \[ \text{no total algorithm decides }\mathcal R(L)\text{ from the order table of }L. \] The source also specifies a nonrepresentable lattice $L_0$ of least cardinality, with $|L_0|>7$ and every smaller finite lattice representable. Its group-interval assertion is \[ \text{no total algorithm decides }\exists G\text{ finite},\ H\le G:\ L\cong[H,G]. \] The undecidability theorem additionally rules out a total computable bound, depending only on $|L|$, on the smallest carrier size among representable lattices.

Outline of the argument

The obstruction is first phrased for subgroup intervals using specially tested lattice chains and coordinate transport. Classification-based reductions control possible factor actions, while competing rank and growth estimates contradict the required long configuration; the subgroup-interval equivalence transfers the obstruction to finite algebras.

Finite colored graphs encode the equivalences permitted by all operations of an algebra. Small witness configurations are assembled into lattice representation constraints, and arithmetic divisor obstructions rule out bounded algebraic realizations; a reduction turns the characterization into the claimed undecidability result.

Formal verification

For family 206, the documented theorem identifies representability with a finite colored-graph witness. Undecidability, nonrepresentation, and subgroup-interval conclusions are not contained in that selected graph equivalence. The listed Comparator files are FiniteCongruenceGraph.lean. The inventory does not supply a named declaration for these Comparator entries. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

Go to contribution →