Papers

We prove that if the product of two class transpositions has finite order, then its order divides 840. This gives a complete affirmative answer to Question 18.48 in the Kourovka Notebook. We also obtain a criterion for the product of two class transpositions to have infinite order. In addition, we construct a pair of class transpositions whose product has order 840, showing that the bound is sharp.

We construct finite-dimensional Lie-group summaries of based rotational paths that retain the exact terminal attitude and selected iterated angular-velocity integrals. These summaries, called rotational trajectory synopses, compose exactly under path concatenation. Building on the classical works of Magnus and Chen, we establish endpoint-retaining body, spatial, and hybrid synopsis groups at every finite order and give a uniform matrix realization for the evaluated Magnus hierarchy. We distinguish this hierarchy from the full Chen signature from order three onward and prove \chg{that the exact endpoint is independent of every finite truncation of the body, spatial, or combined angular-velocity signature: the synopsis maps are surjective at every finite order}. An underactuated satellite example connects the retained coordinates to geometric phase in spacecraft attitude dynamics.

Given a prime $p$, the Nottingham group $G(p)$ is the group of formal power series $x+a_2x^2+a_3x^3+\cdots$, $a_i\in{\bf Z}/p{\bf Z}$, under substitution. For $n\geq 1$, we write $G_n(p)$ for the quotient of $G(p)$ by its normal subgroup $K_n=\{x+a_{n+1}x^{n+1}+a_{n+2}x^{n+2}+\cdots\,|\, a_i\in {\bf Z}/p{\bf Z}\}$, namely the group of truncated polynomials $x+a_2x^2+\cdots+a_nx^n$, $a_i\in {\bf Z}/p{\bf Z}$, under substitution, which is a $p$-group of order $p^{n-1}$ generated by $x+x^2,\dots,x+x^n$. In this paper, we determine the power-commutator presentation of $G_{15}(p)$ relative to $x+x^2,\dots,x+x^{15}$. This, in turn, automatically gives the power-commutator presentation of $G_{n}(p)$ relative to $x+x^2,\dots,x+x^{n}$ for any $1\leq n<15$, as well as the initial segments of the power and commutator relations in $G_{n}(p)$ corresponding to $x+x^2,\dots,x+x^{15}$ for any $n>15$.

We investigate finite equational bases for full power semirings of finite semigroups, including the empty set, in the constant-free signature with addition and multiplication. For a nontrivial finite nilpotent semigroup, we associate a finite relational structure recording the ordered products that are nonzero. We prove that the full power semiring is finitely based if and only if this structure has finite duality, and relate this condition to first-order definability and dismantling of the square of its core. Quantitative bounds connect obstruction size with the number of variables required in an identity basis. In the commutative case of nilpotency index $d$, the criterion reduces to the existence of an element with nonzero $(d-1)$st power. We also establish a nonfinite-basis obstruction for semigroups with a two-element group ideal and prove a finite lifting theorem. These results yield a direct-product criterion and a five-element counterexample to sufficiency of the identity-fibre condition. The arguments use equational logic, finite relational duality, and explicit algebraic constructions.

We prove that every finite abelian group arises as the Schur multiplier of a finite solvable group, from which it can be deduced that every finite abelian group is the knot group of some extension of number fields.

We study tilings of symmetric and alternating groups by conjugacy classes. For $S_n$, the case where the conjugacy class consists of transpositions was first investigated by Rothaus and Thompson in 1966 and remains open. We prove that this long-standing case is the only unresolved case for symmetric groups: every other nonidentity conjugacy class of $S_n$ fails to tile $S_n$. Our proofs combine representation-theoretic methods with combinatorial arguments concerning products of permutations. We also completely resolve the corresponding problem for alternating groups in a stronger form: for $n\geq5$, no normal subset of $A_n$ that avoids the identity can tile $A_n$. \iffalse We study the problem of whether a conjugacy class tiles the symmetric group or alternating group. For $S_n$, the case where the conjugacy class consists of transpositions was first investigated by Rothaus and Thompson in 1966 and remains open. We prove that this long-standing case is the only unresolved case for symmetric groups: every other nonidentity conjugacy class of $S_n$ fails to tile $S_n$. The proof combines representation-theoretic methods with combinatorial arguments concerning products of permutations. We also completely resolve the corresponding problem for alternating groups in a stronger form: for $n\geq5$, no normal subset of $A_n$ that avoids the identity can tile $A_n$. \fi \iffalse In this paper, we study when a conjugacy class or a normal subset tiles the symmetric or alternating group. The case of transpositions was first investigated by Rothaus and Thompson, and in general, it remains open. We show that every other conjugacy class of a symmetric group does not tile. We also prove that normal subsets in $A_n$ ($n\ge 5)$ not containing the identity element cannot tile.

We prove that at every infinite cardinality, every group which is neither abelian of exponent greater than two nor generalized dicyclic admits a graphical regular representation, settling the infinite-group part of Watkins's conjecture. We also determine the Cayley index of every infinite group: it is $1$, $2$, or $8$, according to its algebraic type, and in every case the index is attained by a connected Cayley graph. For every infinite group $G$ of cardinality $\kappa$, we construct $2^\kappa$ pairwise nonisomorphic Cayley graphs with exactly the unavoidable inverse-pair symmetries, diameter two, and $\kappa$ common neighbors at every distinct pair. The principal tool recovers a continuous ordinal hierarchy from an alternating adjacency baseline with bounded-degree errors: robust finite patterns identify the initial classes, successive twin quotients recover the layers, and their finite exception packets determine the translation action. The reconstruction applies without a group action and is stable under additional layerwise bounded-degree edits. Further results give closed Cantor-cube families with prescribed finite data in the regular cases, sharp cofinality-dependent graph properties, and optimal three-valued shortest-path metrics.

The finite group isomorphism problem asks whether two finite groups of order $N$ are isomorphic. The first algorithm, attributed to Tarjan (see Miller, STOC '78), runs in time $N^{\log N + O(1)}$. Despite intensive study, the current best known algorithm has a running time of $N^{(1 / 4 + o(1))\log N}$ (Rosenbaum, '13). $p$-groups of class $2$ have been recognized as the major bottleneck for faster group isomorphism. Recent progress has led to $N^{o(\log N)}$-time algorithms for $p$-groups of class $2$ where $p$ is odd (Sun, STOC '23; Ivanyos–Mendoza–Qiao–Sun–Zhang, FOCS '24; Grochow–Qiao–Stange–Sun, STOC '25). However, the case of $p=2$, which represents the majority of $p$-groups of class 2 assuming a well-known conjecture in group enumeration, remained elusive, with essentially no progress until now. In this paper, we present an algorithm for testing the isomorphism of two 2-groups of Frattini class 2 of order $N$ in time $N^{O((\log N)^{1/2})}$. To our knowledge, this is the first $N^{o(\log N)}$-time isomorphism algorithm for a class of $2$-groups that constitutes logarithmically almost all $2$-groups, in the sense that $\lim_{N \to \infty} \frac{\log(\text{\# 2-groups of Frattini class 2 and order } \leq N)}{\log(\text{\# 2-groups of order} \leq N)} = 1$. As our main tool, we present the first non-trivial algorithms for the quadratic form space/tuple isometry problems over $\mathbb{F}_2$. These algorithms rely on combinations of combinatorial and algebraic ideas, including finite matrix group algorithms developed by Luks (FOCS '92). As far as we know, this is the first time that matrix group algorithms are used to make progress on the worst-case complexity of $p$-group isomorphism.

We prove that if $F$ is a noncyclic free group, $R\triangleleft F$ and $S\le F$, then the inclusion $R'\le S'$ implies $R\le S$. Thus the normality hypothesis on the second subgroup can be omitted from the Auslander–Lyndon theorem. No restrictions are imposed on ranks or indices.

We obtain a characterization of finite non-$p$-groups with five automorphism orbits. In the solvable case of Fitting height $3$, the groups are precisely $S_4$ and $C_2^4\rtimes D_{10}$. Moreover, the corresponding fixed-point subspaces form a spread in a finite projective space. For Fitting height $2$, we distinguish the possible structures according to the number of prime divisors of $|G|$.

Let $\Gamma<\mathrm{Isom}(\mathbb H^n)$, $n\geq2$, be a cocompact lattice containing a convex cocompact codimension-one subgroup in the sense of Sageev. We prove that every finitely supported admissible random walk on $\Gamma$ has hitting measure singular with respect to Lebesgue measure on $\partial\mathbb H^n$. The proof extends the method of Kosenko–Tiozzo, replacing Fourier analysis for a cyclic subgroup by a von Neumann dimension argument that also applies to nonabelian subgroups. As corollaries, we prove the singularity conjecture of Kaimanovich and Le Prince for cocompact hyperbolic lattices admitting a proper cocompact cubulation, cocompact Kleinian groups, and cocompact hyperbolic lattices with totally geodesic sublattices of codimension one. In particular, our result covers all cocompact hyperbolic reflection groups, cocompact arithmetic lattices of simplest type and cocompact nonarithmetic lattices arising from the hybrid and inbreeding constructions.

In algorithmic statistics a string x is explained by a finite set containing it, and Kolmogorov's structure function records the smallest such model at each level of complexity. Strong models, those computable from the data by a total algorithm, are essentially the cells of simple partitions, so a partition of {0,1}^n can be read as a hypothesis and the cell containing x as the model it assigns. We develop algorithmic statistics over symmetric partitions, the orbit partitions of groups acting on strings. The Galois connection between subgroups and partitions gives each ambient group G a lattice of symmetric partitions with canonical certificates, joins and meets. A set is a cell of a symmetric partition exactly when its setwise stabiliser in G acts transitively on it, so the resulting structure function is Kolmogorov's restricted to these G-homogeneous sets. With a symmetric sophistication, it measures which part of the regularity of x is symmetric. Under the full symmetric group, cells recover all models and cells of cheap partitions recover exactly the strong models, so normal and strange strings are characterised by symmetry. For GL(n,2) the homogeneous sets are the linearly homogeneous ones, whose XOR dependencies look the same from every point. The GL structure function of a nonzero x lies in a band between C(x) - alpha and n - alpha, and both edges are attained: some stochastic normal strings have simple structure invisible to linear symmetry, with sophistication near 0 but GL-sophistication near C(x). We coordinatise permutation groups by a Burnside ring element (type) and a permutation (placement). In these coordinates a linear hypothesis is determined by its type up to n^2 bits, and any space of symmetry hypotheses small enough to search is small enough to miss simple structure. This sets up learned search over orbit models, the subject of later papers in the series.

In this paper, we present a polynomial-time quantum algorithm for computing a minimum-sized generating set of solvable black-box groups. Next, we consider the class $\Gamma_d$ of black-box groups, where every non-abelian composition factor is isomorphic to a subgroup of the symmetric group $S_d$ for a fixed $d$. We design polynomial-time quantum algorithms to compute the direct product decomposition of abelian factor groups and solve the constructive membership problem for factor groups of groups from $\Gamma_d$. With the help of these algorithms, we design a quantum algorithm for computing a chief series of black-box groups from $\Gamma_d$. Using the chief series, we construct a polynomial-time quantum algorithm for computing minimum generating sets of black-box groups from $\Gamma_d$. Finally, we show that the minimum generating set problem for general black-box groups is in $\textrm{NP} \cap \textrm{coAM}$.

We call a countable discrete group cocycle-hyperlinear if all its twisted group von Neumann algebras are Connes-embeddable. We show that the class of cocycle-hyperlinear groups includes extensions of treeable groups by amenable groups and central extensions of SL(n,Z), GL(n,Z) and PSL(n,Z) for n greater than 1. We also show that this class is closed under taking direct products, free products, amalgamated free products with amenable amalgams, HNN extensions by amenable groups, as well as certain quotients, extensions, and overgroups.

We study the finite basis problem for flat semirings of the forms \(S_c(W)\), \(M_c(W)\), and \(M(W)\), where \(W\) is a nonempty set of words in a free commutative semigroup, a free commutative monoid, and a free monoid, respectively. We completely classify such flat semirings with respect to the finite basis property, allowing \(W\) to be infinite. We prove that \(S_c(W)\) is finitely based if and only if every word in \(W\) is either a cube of a letter or has length at most two, whereas \(M_c(W)\) and \(M(W)\) are finitely based if and only if \(W\) consists solely of the empty word. As applications, we recover the nonfinite basability of \(\flat(\mathbb{Z})\) and the max-plus semiring \((\mathbb{Z},\max,+)\).

The finite basis problem for small semirings differs from its semigroup counterpart even in order three. Recent work classifies several additive types of four-element additively idempotent semirings, including all 386 semirings whose additive reduct is a chain. We consider all four-element semirings with commutative addition, in the binary signature without named constants. Combining the existing classifications with structural reductions and polynomial normal forms, we obtain 2284 finitely based and 57 nonfinitely based isomorphism types among the 2341 types. The nonfinitely based cases consist of 45 additively idempotent semirings and twelve with nonidempotent addition. The positive arguments give explicit bases or finite constructions with specified bounds. The negative arguments use cited results, term retractions and a hypergraph obstruction. A complete catalogue records the applicable result for each representative; separate correspondence tables identify precisely the cases supplied by the earlier classifications.

Analogues of the Sylow, Hall and Schur-Zassenhaus theorems for finite groups have recently been generalised to finite skew braces, providing a powerful framework for their structural analysis. Building on this foundation, the natural next step of skew braces is the study of Carter sub-skew braces, namely, centrally nilpotent, self-idealising sub-skew braces. However, while every finite soluble group possesses Carter subgroups, not every finite soluble skew brace possesses Carter sub-skew braces. We introduce Carter sub-skew braces, analyse their structural limitations and establish Carter-like theorems. We prove that finite soluble skew braces have Carter sub-skew braces if every sub-skew brace of prime-power order is a centrally nilpotent left ideal. We also prove their existence within finite multipermutational skew braces.

We prove two new cases of the Stuck–Zimmer conjecture. First, we settle the Stuck–Zimmer conjecture for ergodic probability preserving actions of irreducible lattices (both uniform and non-uniform) in connected semisimple real Lie groups with finite center, no compact factor and real rank at least two. We furthermore extend the Stuck–Zimmer theorem to all irreducible probability preserving actions of such groups whenever one simple factor is locally isomorphic to $\mathrm{SL}_2(\mathbb R)$. In particular, this settles the Stuck–Zimmer conjecture for irreducible actions of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$. The proof relies on a new stabilizer rigidity criterion using local spectral gap of projected stabilizers in a single simple factor. Here local spectral gap is understood in the sense of Boutonnet, Ioana and Salehi Golsefidy.

We investigate the structural, combinatorial, ideal-theoretic and Krohn-Rhodes complexity of the Margolis-Rhodes monoid, MR(G), of a finite simple graph G, viewed topologically as a 1-dimensional simplicial complex. Alongside the full monoid, we examine some subsemigroups including St(G), defined by the condition that the full inverse image is an edge or the empty set and Inj(G), the monoid of all partial 1-1 continuous functions. We provide explicit combinatorial enumerations and struture for paths and cycles. We compute Green's relations showing in particular that the partial order of regular J-classes is isomorphic to the poset of induced subgraphs of G. Finally, we apply these structural invariants to Krohn-Rhodes complexity theory. It is known that the Margolis-Rhodes monoid has complexity at most 2 and 1 for St(G) and Inj(G). We show that for cycles the complexity of its Margolis-Rhodes monoid is 2 if and only if the cycle is of length at least 4. For paths, we prove that the complexity of its Margolis-Rhodes monoid is 2 if the path length is at least 13.

Let \( G \) be a finite group, and consider its common non-zero graph, which we denote by \( \Gamma_{nv}(G) \). In this graph, the vertices represent the non-linear irreducible characters of \( G \). There is an edge connecting two distinct vertices \( \chi_1 \) and \( \chi_2 \) if there is a vanishing element \( g \in G \) such that \( \chi_1(g) \chi_2(g) \neq 0 \). A finite group \( G \) is described as a strongly \( \mathcal{H}_1' \)-group if the set of vanishing elements of each of its non-linear irreducible character is equal to the set of vanishing elements of \( G \). In this paper, we prove Conjecture 1 from \cite{ourself}, which states that if \( \Gamma_{nv}(G) \) is null (has no edge), then \( G \) must either be a strongly \( \mathcal{H}_1' \)-group or a Frobenius group, whose Frobenius complement is isomorphic to \( Q_8 \). Additionally, we show that if there are no triangles in \( \Gamma_{nv}(G) \), this implies that the group is solvable. We also look at the situation where the common non-zero graph of a finite group \( G \) contains only one edge. This leads us to conclude that \( G \) is either isomorphic to \( S_4 \); or that the single edge in \( \Gamma_{nv}(G) \) is of the form \( \{\theta, \zeta\theta\} \) for some \( \zeta \in \Irr(G/G') \) and \( \theta \in \Irr(G|G') \), $G$ is a $p$-nilpotent group for some prime $p$ and for all characters \( \chi \in \Irr(G|G') - \{\theta, \zeta\theta\} \), we find \( \Van(\chi) = \Van(G) \). Lastly, our study also explores groups whose common non-zero graph forms a star shape, concluding that such configurations can only occur if \( G \) has exactly one non-linear irreducible character.

Let $\Sigma_{1} = \mathbb{N}^*$ and $\Sigma_{k+1} = \overline{\mathbb{N}^* + \Sigma_{k}}$ be the cumulative depth filtration of $\beta\mathbb{N}$, the analogue for $(\mathbb{N},+)$ of a chain of closed ideals that Protasov and Protasova studied for discrete groups, where strict descent follows from a theorem of Lutsenko and Protasov. For every $k$ we give an explicit set whose closure meets $\Sigma_{k}$ but not $\Sigma_{k+1}$. Fix the doubly exponential sequence $e_{n} = 2^{2^{n}}$, partition it into $k$ subsequences $E_{0}, \dots, E_{k-1}$ by the residue of the index modulo $k$, and set $A_{k} = E_{0} + \cdots + E_{k-1}$. We prove that any sum $q_{0} + \cdots + q_{k-1}$ of free ultrafilters with $E_{t} \in q_{t}$ lies in $\Sigma_{k} \setminus \Sigma_{k+1}$. The engine is a master lemma, proved by induction on $j$: if a sum $F_{1} + \cdots + F_{j}$ of subsequences of $\{e_{n}\}$ with pairwise disjoint index sets belongs to a free ultrafilter $s$, then $s \notin \Sigma_{j+1}$. The proof rests on a single rigidity of the doubly exponential sequence: a fixed difference forces the largest index in any shift-intersection, once it is large, to cancel within its own subsequence, which makes every shift-intersection descend by at least one level. The same witnesses lie in the gaps of the pure filtration.

We show that cohomological and homological goodness are equivalent. We prove that if $G$ is a coherent even Artin group or an even Artin group whose underlying graph does not contain 4-clique or a LERF Artin group then $G \otimes G$ and $G \wedge G$ are residually finite. Furthermore for $G$ a RAAG both $G \otimes G$ and $G \wedge G$ are residually $p$-finite for any prime integer $p$. For $G$ a limit group or a Coxeter group we show that $G \wedge G$ is residually finite. For $G$ a soluble group of type $FP_{\infty}$ we prove that the groups $\X(G)$, $\nu(G)$, $G \wedge G$ are virtually residually $p$-finite for almost all primes $p$. For $G$ a generalised Baumslag-Solitar group of rank $n \geq 1$ that is residually finite, we show that $G \wedge G$ is residually finite. For $G = SL_n(\Z)$ and $G = GL_n( \mathbb{Z})$, $n \geq 3$, we show that $G \wedge G$ is residually finite.

We show that there exists a finitely generated linear group whose non-abelian tensor square is not linear. On the other hand, the non-abelian tensor square of the braid group $B_n$, $n>3$, is linear.

Let $X$ be a connected finite-type CW-complex with fundamental group $G$. We show that the profinite completion $\widehat{G}$ determines the cohomology jump loci $\mathcal{V}^q_s(X,\mathbb{C})$ under two hypotheses: that the loci are finite unions of torsion-translated subtori, as for smooth quasi-projective varieties, and that $\widehat{G}$ determines the Betti numbers of the finite cyclic covers of $X$ in degrees $\le q$, which holds unconditionally for $q=1$, and in all degrees when $X$ is aspherical and $G$ is good in the sense of Serre. When the isomorphism of completions is compatible with an identification of the abelianizations, the loci correspond exactly; in general, they correspond up to an isogeny. Applied to finite covers, this shows that the tropical bounds for the Bieri–Neumann–Strebel–Renz invariants are profinite invariants, and recovers the profinite invariance of the BNS invariant of Kähler groups due to Hughes, Llosa Isenrich, Py, Stover, and Vidussi. We also show that $\widehat{G}$ determines the graded abelian groups $\mathrm{gr}_r(G/W(G))$, torsion included, for every verbal subgroup $W(G)$; the cases $W(G)=1$ and $W(G)=G''$ give the lower central series quotients and the Chen groups. For hyperplane arrangements, it follows that no arithmetic Zariski pair is distinguished by any of these invariants, while two known lattice-isomorphic pairs show, respectively, that the profinite completion of an arrangement group is not combinatorially determined, and that it does not determine the group.

Let $B_\infty$ and $P_\infty$ be the braid group and the pure braid group on infinitely many strands. We show that the inclusion $C^*_r(P_\infty)\subseteq C^*_r(B_\infty)$ is $\mathcal{Z}$-stable and selfless. Hence every intermediate inclusion and every intermediate $C^*$-algebra is $\mathcal{Z}$-stable and selfless. The intermediate $C^*$-algebras are natural examples of simple nonnuclear $\mathcal{Z}$-stable $C^*$-algebras, and they are singly generated. The proof of $\mathcal{Z}$-stability uses a simple relative centralizer criterion. We also show that $C^*_r(\mathbb{F}_\infty)\subseteq C^*_r(B_\infty)$ is a selfless inclusion, where $\mathbb{F}_\infty$ is the free group of braids in which only the first string moves. All results hold for reduced twisted group $C^*$-algebras. Selflessness of $C^*_r(B_\infty)$ itself already follows from work of Ozawa. On the other hand, $\mathcal{Z}$-stability fails for all twisted group $C^*$-algebras of groups that are not inner amenable. A variant shows that it fails for all twisted group $C^*$-algebras of $B_n$ with $n$ finite, and for the twisted crossed products from Artin's representation of $B_n$. Finally the method applies to other increasing unions of groups, such as $\mathrm{SL}_\infty(\mathbb{Z})$, whose reduced group $C^*$-algebra is $\mathcal{Z}$-stable.

Based on a recent generalization of the Reidemeister spectrum of a finite group [1], in this short note we extend the main result of our previous paper [2].

Let $A$ be a finite-dimensional local algebra over a splitting field. We prove that an irreducible tensor factorization $A \cong P_1 \otimes \dotsb \otimes P_n$ of the algebra $A$ is unique up to isomorphism and order of $P_1, \dotsc, P_n$.

A skew-morphism of a finite group $G$ is a permutation $\sigma$ of $G$ fixing the identity element such that there exists a function $\pi\colon G\to\mathbb{Z}$ satisfying $\sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y)$ for all $x,y\in G$. For a given skew-morphism $\sigma$ of $G$, the product of the left regular representation of $G$ and the cyclic group $\langle\sigma\rangle$ forms a permutation group on $G$, called a skew-product group of $G$. Skew-product groups of finite elementary abelian $p$-groups were investigated in \cite{DLYZ}. In the present paper, we extend this study to all finite abelian $p$-groups.

Lazard asked which $N$-series of a group arise from multiplicative filtrations of the integral group ring. Given an $N$-series $H_\bullet$ of a group $G$, let $A_\bullet$ be the filtration of the augmentation ideal spanned by products of elements $x-1$, $x\in H_a$, weighted by $a$. Every realizing filtration contains $A_\bullet$, so $H_\bullet$ is realizable exactly when the maps $H_n/H_{n+1}\to A_n/A_{n+1}$, $h\mapsto h-1$, are injective. We show that each kernel is the cokernel of an explicit map of normalized bar groups; for a finite group this makes realizability decidable by integer linear algebra. For a profinite group with a cofinal tower of finite quotients and an $N$-series induced from it, suppose $h\in H_n$ and $h-1\in A_{n+1}$ in every finite quotient. Then $h-1\in A_{n+1}$ in the discrete group ring exactly when the widths of its finite-level Losey expressions are bounded. This descent can fail. Using Tahara's class-three $2$-groups and the Hartl-Mikhailov-Passi description of the fourth dimension subgroup, we construct a countable product $P$ of finite $2$-groups. It contains an element of $\gamma_3(P)$ that lies in the fourth dimension subgroup of every finite subproduct of $P$ but not in $D_4(P)$.

We study g-dimonoids, which are algebraic structures equipped with two associative binary operations satisfying a prescribed system of axioms. We develop a construction of g-dimonoids based on inflations of semigroups, construct several classes of 2-dinilpotent commutative g-dimonoids, and investigate the isomorphism problem for these classes. As an application of these results, we obtain a structural description of all commutative g-dimonoids of order 3, up to isomorphism.