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2025-12-04
Ushakov, Alexander, Ushakov, Alexander
Regardless of the choice of parameters, knowledge of a single signed message, i.e., a pair message/signature, produced by Kahrobaei-Koupparis digital signature scheme, proposed in [D. Kahrobaei and C. Koupparis, 2012], is sufficient to forge a valid signature for any other message.
DOI: 10.46298/jgcc.2025.17.2.169952025-12-04
Gaglione, Anthony M., Spellman, Dennis, Gaglione, Anthony M., Spellman, Dennis
Let $p$ and $n$ be positive integers. Assume additionally that $p\neq 3$ is a prime and that $n>2$. Let $R$ be a field of characteristic $p$. A very special consequence of a result of Bunina and Kunyavskii (2023, arXiv:2308.10076) is that $SL_{n}(R)$ is co-Hopfian as a group if and only if $R$ is co-Hopfian as a ring. In this paper, we prove that if $k$ is the algebraic closure of the $2$ element field, then $SL_{2}(k)$ is a co-Hopfian group. Since this $k$ is trivially seen to be co-Hopfian as a ring our result somewhat extends that of Bunina and Kunyavskii. We apply our result to prove that the class of groups satisfying Turner's Retract Theorem (called Turner groups here) is not closed under elementary equivalence thereby answering a question posed by the authors in (2017, Comm. Algebra).
DOI: 10.46298/jgcc.2025.17.2.168752025-05-26
Crowe, Gemma, Crowe, Gemma
In this paper we provide an alternative solution to a result by Juhász that the twisted conjugacy problem for odd dihedral Artin groups is solvable, that is, groups with presentation $G(m) = \langle a,b \; | \; _{m}(a,b) = {}_{m}(b,a) \rangle$, where $m\geq 3$ is odd, and $_{m}(a,b)$ is the word $abab \dots$ of length $m$, is solvable. Our solution provides an implementable linear time algorithm, by considering an alternative group presentation to that of a torus knot group, and working with geodesic normal forms. An application of this result is that the conjugacy problem is solvable in extensions of odd dihedral Artin groups.
DOI: 10.46298/jgcc.2025.17.1.135612025-02-24
Dermenjian, Aram, Evetts, Alex, Dermenjian, Aram, Evetts, Alex
We study the conjugacy class growth function in finitely generated virtually abelian groups. That is, the number of elements in the ball of radius $n$ in the Cayley graph which intersect a fixed conjugacy class. In the class of virtually abelian groups, we prove that this function is always asymptotically equivalent to a polynomial. Furthermore, we show that in any affine Coxeter group, the degree of polynomial growth of a conjugacy class is equivalent to the reflection length of any element of that class.
DOI: 10.46298/jgcc.2025.17.1.134592024-07-17
Buchinskiy, Ivan, Kotov, Matvei, Treier, Alexander, Buchinskiy, Ivan, Kotov, Matvei, Treier, Alexander
In this paper, we prove a criterion for a predicate structure to be equationally Noetherian.
DOI: 10.46298/jgcc.2024.16.1.138722024-05-14
Gvozdevsky, Pavel, Gvozdevsky, Pavel
We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with countable underlying sets: totally ordered sets, fields, and groups. This answers an old question by B. Plotkin for groups.
DOI: 10.46298/jgcc.2024.16.1.134932023-09-28
Ferov, Michal, Pengitore, Mark, Ferov, Michal, Pengitore, Mark
In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian.
DOI: 10.46298/jgcc.2023.15.1.117282023-02-04
Marjanski, Lindsay, Solon, Vincent, Zheng, Frank, Zopff, Kathleen, Marjanski, Lindsay, Solon, Vincent, Zheng, Frank, Zopff, Kathleen
In this paper, we study geodesic growth of numbered graph products; these are a generalization of right-angled Coxeter groups, defined as graph products of finite cyclic groups. We first define a graph-theoretic condition called link-regularity, as well as a natural equivalence amongst link-regular numbered graphs, and show that numbered graph products associated to link-regular numbered graphs must have the same geodesic growth series. Next, we derive a formula for the geodesic growth of right-angled Coxeter groups associated to link-regular graphs. Finally, we find a system of equations that can be used to solve for the geodesic growth of numbered graph products corresponding to link-regular numbered graphs that contain no triangles and have constant vertex numbering.
DOI: 10.46298/jgcc.2023.14.2.100192022-10-04
Levine, Alex, Levine, Alex
We give an algorithm that decides whether a single equation in a group that is virtually a class $2$ nilpotent group with a virtually cyclic commutator subgroup, such as the Heisenberg group, admits a solution. This generalises the work of Duchin, Liang and Shapiro to finite extensions.
DOI: 10.46298/jgcc.2022.14.1.97762022-08-11
Grochow, Joshua A., Qiao, Youming, Tang, Gang, Grochow, Joshua A., Qiao, Youming, Tang, Gang
We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in \mathbb{F}_q[x_1,\dots, x_n]$, and decides whether $f$ and $g$ are isomorphic in time $q^{O(n)}$ for most $f$. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow-Qiao, ITCS, 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.
DOI: 10.46298/jgcc.2022.14.1.94312021-11-11
Sahattchieve, Jordan A., Sahattchieve, Jordan A.
An erratum to this article is posted at https://gcc.episciences.org/page/errata This paper generalizes results of M. Moon on the fibering of certain compact 3-manifolds over the circle. It also generalizes a theorem of H. B. Griffiths on the fibering of certain 2-manifolds over the circle.
DOI: 10.46298/jgcc.2021.13.2.70722021-04-15
Mijares, Sebastià, Ventura, Enric, Mijares, Sebastià, Ventura, Enric
An extension of subgroups $H\leqslant K\leqslant F_A$ of the free group of rank $|A|=r\geqslant 2$ is called onto when, for every ambient free basis $A'$, the Stallings graph $\Gamma_{A'}(K)$ is a quotient of $\Gamma_{A'}(H)$. Algebraic extensions are onto and the converse implication was conjectured by Miasnikov-Ventura-Weil, and resolved in the negative, first by Parzanchevski-Puder for rank $r=2$, and recently by Kolodner for general rank. In this note we study properties of this new type of extension among free groups (as well as the fully onto variant), and investigate their corresponding closure operators. Interestingly, the natural attempt for a dual notion -- into extensions -- becomes trivial, making a Takahasi type theorem not possible in this setting.
DOI: 10.46298/jgcc.2021.13.1.70362020-06-24
Nikolaev, Andrey, Ushakov, Alexander, Nikolaev, Andrey, Ushakov, Alexander
We consider a group-theoretic analogue of the classic subset sum problem. In this brief note, we show that the subset sum problem is NP-complete in the first Grigorchuk group. More generally, we show NP-hardness of that problem in weakly regular branch groups, which implies NP-completeness if the group is, in addition, contracting.
DOI: 10.46298/jgcc.2020.12.1.6541