We give an elementary affirmative answer to Problem~20.73 of the
Kourovka Notebook. More precisely, we show that every countably
infinite group admits a factorization into two infinite subsets in
which each element has a unique representation. The main ingredient
is an inductive extension lemma for groups in which every nonidentity
element has only finitely many square roots. The general case follows
by combining this lemma with elementary reductions involving abelian
subgroups, infinite-index subgroups, and the FC-centre.
2 green6 unreviewed1 full-paper AI check
PF:2026.000017ProofForum Algebra & Number Theory Group Theory v2
We prove that every torsion-free locally graded group in which each
subgroup is either pronormal or contained in the pro-norm is abelian.
Consequently, no infinite simple locally graded group has this
property. We also extend the classification of insoluble locally
$c$-graded groups with the same property to all locally graded
groups. The proof combines results on pronormal subgroups with the
theory of finitely generated HNN-free groups.
1 green5 unreviewed1 full-paper AI check
PF:2026.000002ProofForum Algebra & Number Theory Group Theory v3
For every prime $p$ we construct a family of balanced finite two-generator $p$-groups whose nilpotency class is arbitrarily large. This gives an affirmative answer to Problem 17.113 in the Kourovka Notebook.
2 yellow1 full-paper AI check
PF:2026.000019ProofForum Algebra & Number Theory Group Theory v1
We answer a question of Brescia, Ersoy and Kuzucuoglu concerning the
subgroup lattice of a $\kappa$-existentially closed group. We prove,
for every infinite cardinal $\kappa$, that such a group contains an
elementary abelian $2$-subgroup of rank $\kappa$. We then construct an
explicit ordered set of cardinality $\kappa$ without endpoints, every
nonempty open interval of which has cardinality $\kappa$. Indexing
a basis of the elementary abelian subgroup by this set gives a strictly
increasing chain of subgroups with exactly $\kappa$ members between
any two distinct members. No regularity assumption on $\kappa$ is
required.
For a graph $F$, the $F$-degree of a vertex counts the copies of $F$ containing it, without requiring the copies to be induced. We determine the eventual regularity spectrum of connected patterns of order at least three: stars admit no regular $F$-irregular hosts, whereas every non-star $F$ admits a connected $r$-regular $F$-irregular host for every sufficiently large integer $r$. These hosts can be chosen to have diameter two and $r+o(r)$ vertices, asymptotically optimal to first order. The construction uses complements of sparse layered regular graphs. Rooted-homomorphism localization and quotient expansion isolate the leading contributions to $F$-degree differences; spectral amplification separates competing terms, and a scale-preserving choice of mixed-radix parameters realizes each prescribed regularity.
19 unreviewed1 full-paper AI check
PF:2026.000016ProofForum Algebra & Number Theory Group Theory v1
For every oracle $X\subseteq\mathbb N$ we construct a three-generated group $G_X$ with an $X$-computably enumerable presentation such that a finitely generated group has an $X$-computably enumerable presentation if and only if it embeds as a finitely generated subgroup of a quotient of a finite free product of copies of $G_X$ by finitely many relators. This gives an affirmative answer to Kourovka Problem 20.33.
2 green1 full-paper AI check
PF:2026.000014ProofForum General & Historical Mathematics General Mathematics v1
An allocation of indivisible goods is MMS-fair if every agent receives at least
her maximin share, and such allocations need not exist.
We classify the complexity of exact maximin shares for additive valuations,
with the values encoded in binary or in unary.
With values in binary, computing a maximin share is $\mathrm{OptP}$-complete
and computing an MMS-fair allocation is $\mathrm{FP}^{\mathrm{NP}}$-complete, already for two agents with identical valuations.
With values in binary, deciding whether an MMS-fair allocation exists is $\mathrm{P}^{\mathrm{NP}}$-complete,
already for four agents holding three distinct valuations.
With values in unary and the number of agents part of the input,
computing a maximin share is $\mathrm{OptP}[O(\log n)]$-complete
and computing an MMS-fair allocation is complete for Chen and Toda's class $\mathrm{FNP}/\!\!/\mathrm{OptP}[O(\log n)]$, again with identical valuations.
In the same setting, deciding existence is $\mathrm{P}^{\mathrm{NP}}[O(\log n)]$-complete.
An MMS-fair allocation always exists if all agents but one share a valuation,
so the two existence results need agents with differing valuations, and the two computation results do not.
All four reductions share one framework, in which a common weight fixes the admissible partitions
and small charges encode a formula, and each uses one of two devices for satisfiability, one for each encoding.
The proofs were found, and this text was written, in interaction with Anthropic's Claude models.
The human author has checked the proofs in the main text, but not yet those in the appendix.
Woronowicz's purification of a factor state on a $C^*$-algebra is characterized by a doubled extension that is pure, $j$-positive, and exact. Exactness is the system--ancilla commutant equality: in the GNS representation of the doubled state, the von Neumann algebra generated by the ancilla copy equals the commutant of the original-system algebra. In his 1972 paper Woronowicz explicitly observed that ``the existence of pure non exact states is not clear.'' We give a concrete construction showing that exactness is independent of purity and $j$-positivity, even after the marginal state is fixed.
The first model uses the finitary symmetric group $G=\operatorname{Sym}_{\mathrm{fin}}(\mathbb N_0)$ and the point stabilizer $K=\{g:g(0)=0\}$. On $\ell^2(G)\otimes\ell^2(\mathbb N_0)$ we construct commuting left and twisted-right representations whose vector state satisfies
\[
\varphi(\bar u_g\otimes u_h)=\mathbf 1_{\{g=h\in K\}}.
\]
The resulting state is pure and $j$-positive, and both marginals are the canonical trace. The projection $\mathbf 1\otimes\ket 0\!\bra 0$ lies in the commutant of the system algebra but not in the ancilla algebra. Thus exactness fails strictly. The same state admits the conditional-expectation formula
\[
\varphi(\bar a\otimes b)=\tau\!\left(E(a)^*E(b)\right),
\qquad E:R\to Q\subsetneq R,
\]
where $R$ is the hyperfinite $II_1$ factor and $Q$ is a proper subfactor. By the uniqueness of the separable hyperfinite $II_1$ factor, the construction restricts to two copies of the qubit UHF algebra. Placing the system copy and the ancilla copy on opposite half-chains realizes the doubled state as a pure state on the two-sided qubit chain. Both copy marginals are the product trace, while the system--ancilla commutant equality fails.
This establishes that exactness is an independent part of Woronowicz's characterization of canonical purification. The qubit-chain realization is spatially inhomogeneous, and its $j$-positive form is degenerate; translation-invariant and strictly $j$-positive variants are separate questions.
8 unreviewed1 full-paper AI check
PF:2026.000012ProofForum Algebra & Number Theory Quantum Algebra v1
We prove quantum Laurent positivity for mutation-acyclic
skew-symmetrizable cluster algebras of arbitrary finite rank.
For every integral compatible quantization with invertible frozen
variables, each cluster variable has a Laurent expansion with
coefficients in $\mathbb N[v^{\pm1}]$ in every seed.
For an indecomposable rigid representation of an acyclic valued quiver,
we identify its Hall shuffle character with the character of a
self-dual simple quiver Hecke module. These modules are real and admit
affinizations; modules corresponding to the summands of a rigid direct
sum strongly commute. A Feigin localization and a comparison of graded
composition multiplicities establish positivity at an arbitrary
target seed.