Nielsen and Soelberg proved that a finite subset A of a torsion-free group whose square A · A contains no uniquely represented element has |A| ≥ 8, and exhibited two groups attaining the bound; the first, here G₁, is a two-generator group that is virtually…
A multiset A of |G| elements of a finite group G is quotient-realizable if A = {b(i)c(i)⁻¹} for two enumerations b, c of G; equivalently, if A = {φ(x)x⁻¹: x ∈ G} for some permutation φ of G. Hall proved in 1952 that for abelian G the obvious necessary condi…
For a finite group G put α(G)=|Aut(G)|/|G|. Problem 21.97 of the Kourovka Notebook asks whether every positive rational is of this form; the answer is yes, by a recent theorem of Sureaux, which also settles the nilpotent case. For abelian groups the answer…
Problem 17.32 of the Kourovka Notebook, proposed by O. V. Bogopolski, asks whether the free group Fₙ obeys the Cayley–Hamilton pattern: if w ∈ Fₙ and φ ∈ Aut Fₙ satisfy genw, φ(w), …, φⁿ(w) = Fₙ, must already genw, φ(w), …, φⁿ⁻¹(w) = Fₙ? Two independent sol…
Let U(3,ℤ/mℤ) be the group of upper unitriangular 3 × 3 matrices over ℤ/mℤ, and let its Cayley graph be taken with respect to the four fundamental-root generators I ± E₁₂, I ± E₂₃. The CayleyPy collaboration (arXiv:2509.19162) computed the diameter of this…
For a group G with a finite generating set S, write e(G,S) = limₙ |B_S(n)|^(1/n) for the exponential growth rate of the word metric, and ξ(G) = {e(G,S): S a finite generating set of G} for the growth spectrum. Fujiwara, Kim and Tanaka have recently determin…
Let Fᵣ be the free group of rank r, let Sₙ be its sphere of radius n — the freely reduced words of length exactly n — and let Bₙ be the corresponding ball. A subset X of a group is independent when no element of X lies in the subgroup generated by the other…
Three recent papers of V. M. Gavrylkiv count, up to isomorphism, the dimonoids, the g-dimonoids and the doppelsemigroups of small order, together with their commutative, abelian, rectangular and strong subclasses, and each closes with one problem asking for…
A doppelsemigroup is a set carrying two associative operations ⊣ and ⊢ that satisfy the two interassociativity laws (x ⊣ y) ⊢ z=x ⊣ (y ⊢ z) and (x ⊢ y) ⊣ z=x ⊢ (y ⊣ z); it is rectangular when both operations obey x · y · z=x · z, in which idempotency is not…
The power semigroup P(S) of a semigroup S is the set of nonempty subsets of S under A · B={ab: a ∈ A, b ∈ B}, and S is globally determined when P(S)congP(T) forces Scong T. Whether every finite semigroup is globally determined is a question of Schein and, i…
A finite subset A of a group is non-UP if no element of A · A is a product ab with a,b ∈ A in exactly one way. In the Promislow group P — the orientable Hantzsche–Wendt Bieberbach group of dimension three, where Promislow found a 14-element non-UP set in 19…
Kinyon and Phillips study loops in which every square lies in two of the three nuclei, and leave three problems open; each of the three asks for a loop with left nuclear squares. We prove that two of them have no counterexample of prime order and none of or…
An involution system is a pair (W,S) with W a group generated by a finite set S of involutions; its growth series is W(q)=Σ_(w ∈ W)q^(ℓ(w)), where ℓ is word length with respect to S. Dos Santos, Hohlweg and Trufanov attach to an even meet involution system…
A tuple (a₁,…,aₙ) of positive integers is G-harmonic if the group G has subgroups U₁,…,Uₙ with [G:Uᵢ]=aᵢ whose left cosets can be translated so as to be pairwise disjoint, and ℤ-harmonic if there are integers r₁,…,rₙ with rᵢ ≢ rⱼ (mod gcd(aᵢ,aⱼ)) for i ≠ j.…
For a finite group G the small Davenport constant d(G) is the greatest length of a sequence over G no non-empty subsequence of which admits an ordering whose product is the identity. Godara and Sarkar conjectured d(H_(p³))=3p-3 for the Heisenberg group of o…
A finite group H is a B-group if every primitive permutation group containing a regular subgroup isomorphic to H is 2-transitive; equivalently, H fails to be a B-group exactly when it occurs as a regular subgroup of a uniprimitive group. Herbig has classifi…
For a balanced two-generator presentation R=⟨ x,y | r₀,r₁⟩ write E(R)=|r₀|+|r₁| for the total length of its freely reduced relators, and call a sequence of elementary Andrews–Curtis moves peak-bounded by B if every presentation it visits, endpoints included…
