International Journal of Algebra and Computation

We determine when two almost automorphisms of a regular tree are conjugate. This is done by combining the classification of conjugacy classes in the automorphism group of a level-homogeneous tree by Gawron, Nekrashevych and Sushchansky and the solution of the conjugacy problem in Thompson’s $V$ by Belk and Matucci. We also analyze the dynamics of a tree almost automorphism as a homeomorphism of the boundary of the tree.

Let $p$ be a prime and let $ℂ$ be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of $\mathrm{\text{GL}}(p,ℂ)$ up to conjugacy. That is, we give a complete and irredundant list of $\mathrm{\text{GL}}(p,ℂ)$-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of $\mathrm{\text{GL}}(p,ℂ)$ is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For $p\le 3$, we classify all finite irreducible subgroups of $\mathrm{\text{GL}}(p,ℂ)$. Our classifications are available publicly in MAGMA.

This paper is devoted to the study of the problem of adjoining a set of bar-units to dimonoids. We give necessary and sufficient conditions for adjoining a set of bar-units to the free left $n$-dinilpotent dimonoid ($n>1$), and prove that it is impossible to adjoin a set of bar-units to the free abelian dimonoid of rank $n>1$ and the free $rs$-dimonoid. As consequences, we establish that it is impossible to extend by a set of bar-units the free left $n$-dinilpotent dimonoid ($n>1$), the free abelian dimonoid of rank $n>1$ and the free $rs$-dimonoid to a generalized digroup. We also count the cardinalities of the free left $n$-dinilpotent dimonoid and the free $rs$-dimonoid for a finite case.

The investigation of co-tame automorphisms of the affine space ${𝔸}_k^n$ is helpful to understand the structure of its automorphisms group. In this paper, we show the co-tameness of several classes of automorphisms, including some 3-parabolic automorphisms, power-linear automorphisms, homogeneous automorphisms in small dimension or small transcendence degree. We also classify all additive-nilpotent automorphisms in dimension four and show that they are co-tame.

The aim of this work is to provide a concrete list of non-isomorphic commutative power-associative superalgebras up to dimension 4 over an algebraically closed field of characteristic prime to 30. As a byproduct, we exhibit an example of a simple non-Jordan power-associative superalgebra whose even part is not semisimple.

For each integer $n\ge 0$, we define a category whose objects are discriminant algebra functors in rank $n$, namely, choices of how to attach functorially to each rank-$n$ algebra a quadratic algebra with the same discriminant. We show that the discriminant algebra functors defined by Loos, Rost, and the present authors are all isomorphic in this category, and prove furthermore that in ranks $n\le 3$ discriminant algebra functors are unique up to unique isomorphism.

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic $\mathrm{\Lambda }$-metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic $\mathrm{\Lambda }$-metric spaces, where $\mathrm{\Lambda }$ is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case $\mathrm{\Lambda }={\mathbb{Z}}^n$ taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic ${\mathbb{Z}}^n$-metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy$:$ $($AMS-$209)$, Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for ${\mathbb{Z}}^n$-free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in ${\mathbb{Z}}^n$, Trans. Amer. Math. Soc. 364 (2012) 2847–2882].

We count the finitely generated subgroups of the modular group $\mathsf{P}\mathsf{S}\mathsf{L}2(\mathbb{Z})$. More precisely, each such subgroup $H$ can be represented by its Stallings graph $\mathrm{\Gamma }(H)$, we consider the number of vertices of $\mathrm{\Gamma }(H)$ to be the size of $H$ and we count the subgroups of size $n$. Since an index $n$ subgroup has size $n$, our results generalize the known results on the enumeration of the finite index subgroups of $\mathsf{P}\mathsf{S}\mathsf{L}2(\mathbb{Z})$. We give asymptotic equivalents for the number of finitely generated subgroups of $\mathsf{P}\mathsf{S}\mathsf{L}2(\mathbb{Z})$, as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size $n$ subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size $n$ subgroup (respectively, finite index subgroup, free subgroup) of $\mathsf{P}\mathsf{S}\mathsf{L}2(\mathbb{Z})$.

If $E$ is a directed graph and $K$ is a field, the Leavitt path algebra $L_K(E)$ of $E$ over $K$ is naturally graded by the group of integers $\mathbb{Z}.$ We formulate properties of the graph $E$ which are equivalent with $L_K(E)$ being a crossed product, a skew group ring, or a group ring with respect to this natural grading. We state this main result so that the algebra properties of $L_K(E)$ are also characterized in terms of the pre-ordered group properties of the Grothendieck $\mathbb{Z}$-group of $L_K(E)$. If $E$ has finitely many vertices, we characterize when $L_K(E)$ is strongly graded in terms of the properties of $K_0^{\mathrm{\Gamma }}(L_K(E)).$ Our proof also provides an alternative to the known proof of the equivalence $L_K(E)$ is strongly graded if and only if $E$ has no sinks for a finite graph $E.$ We also show that, if unital, the algebra $L_K(E)$ is strongly graded and graded unit-regular if and only if $L_K(E)$ is a crossed product.

In the process of showing the main result, we obtain conditions on a group $\mathrm{\Gamma }$ and a $\mathrm{\Gamma }$-graded division ring $K$ equivalent with the requirements that a $\mathrm{\Gamma }$-graded matrix ring $R$ over $K$ is strongly graded, a crossed product, a skew group ring, or a group ring. We characterize these properties also in terms of the action of the group $\mathrm{\Gamma }$ on the Grothendieck $\mathrm{\Gamma }$-group $K_0^{\mathrm{\Gamma }}(R).$