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.
8 unreviewed
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.
6 unreviewed
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 unreviewed
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.
4 unreviewed
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 unreviewed
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.
17 unreviewed
PF:2026.000009ProofForum Algebra & Number Theory Group Theory v1
We prove an integrated computability matrix for exceptional Brunnian words in finite marked group presentations. The matrix separates common-representative literal deletion from quotient killing, arbitrary markings from genuine finite sets, and supplied-word recognition from positive-only and total witness search. We also determine the exact arithmetical upper boundary, identify a sharp one-relator obstruction, and use a sphere benchmark to show that the two Brunnian numerators are genuinely different.
10 unreviewed
PF:2026.000007ProofForum Algebra & Number Theory Group Theory v1
For every order $n\geq3$ we construct an irreducible elementary net of modules over a principal ideal domain in an algebraic extension of its fraction field that is not closed. The example is defined over $\mathbb Z$ inside the cubic field generated by a root of $x^3-x-1$. This gives a negative answer to Kourovka Problem 19.48.
3 unreviewed
PF:2026.000006ProofForum Algebra & Number Theory Group Theory v1
We answer both parts of Kourovka Problem 19.9. First, for $r\geq4$ the finitely generated linear group $\operatorname{Sp}_{2r}(\mathbb Z)$ has a non-linear non-abelian tensor square, by combining the Brown--Loday description for perfect groups with Deligne's non-residually finite universal central extension. Second, for every $n>3$ the tensor square $B_n\otimes B_n$ of the braid group is linear. The latter follows from the homology map $H_2(B_n;\mathbb Z)\to H_2(S_n;\mathbb Z)$ and the standard splitting of the tensor square when the abelianization has no $2$-torsion.