For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $ρ$ and $λ$ giving a (nonassociative) multiplication $x\cdot y=x^ρ+y^λ$. If $S$ is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional $S$-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for $\mathbb{Z}$-linear quasigroups. We consider the extent to which ordinary characters classify $\mathbb{Z}$-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic $\mathbb{Z}$-linear quasigroups with the same ordinary character. For a subclass of $\mathbb{Z}$-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on $\mathbb{Z}$-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy.
In this paper the number of isomorphism classes of the right transversals of the symmetric group $Σ_{n-1}$ in the symmetric group $Σ_n$ is calculated. This result together with the techniques employed along the paper may be used to go forward in the study and classification of the subgroups with a certain number of isomorphism classes of transversals in the group.
We enumerate central and medial quasigroups of order less than $128$ up to isomorphism, with the exception of those quasigroups that are isotopic to $C_4\times C_2^4$, $C_2^6$, $C_3^4$ or $C_5^3$. We give an explicit formula for the number of quasigroups that are affine over a finite cyclic group.
Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity $(x*y)*(x*z)=(y*y)*(y*z)$. Using combinatorial properties of the Cayley kernel and the squaring mapping, we prove that a twisted Ward left quasigroup of prime order is either permutational or a quasigroup. Up to isomorphism, all twisted Ward quasigroups $(X,*)$ are obtained by twisting the left division operation in groups (that is, they are of the form $x*y=ψ(x^{-1}y)$ for a group $(X,\cdot)$ and its automor…
In this paper we describe a family of isomorphism invariants of a finitely generated Coxeter group W. Each of these invariants is the isomorphism type of a quotient group W/N of W by a characteristic subgroup N. The virtue of these invariants is that W/N is also a Coxeter group. For some of these invariants, the isomorphism problem of W/N is solved and so we obtain isomorphism invariants that can be effectively used to distinguish isomorphism types of finitely generated Coxeter groups.
We investigate the relationship between various isomorphism invariants for finite groups. Specifically, we use the Weisfeiler-Leman dimension (WL) to characterize, compare and quantify the effectiveness and complexity of invariants for group isomorphism. It turns out that a surprising number of invariants and characteristic subgroups that are classic to group theory can be detected and identified by a low dimensional Weisfeiler-Leman algorithm. These include the center, the inner automorphism group, the commutator subgroup and the derived series, the abelian radical, the solvable radical, the …
Let $L$ and $H$ be finite-dimensional restricted Lie algebras over a perfect field $\F$ such that $u(L)\cong u(H)$, where $u(L)$ is the restricted enveloping algebra of $L$. We prove that if $L$ is $p$-nilpotent and abelian, then $L\cong H$. We deduce that if $L$ is abelian and $\F$ is algebraically closed, then $L\cong H$. We use these results to prove the main result of this paper stating that if $L$ is $p$-nilpotent, then $L/L'^p+γ_3(L)\cong H/H'^p+γ_3(H)$.
Automorphisms of order $2$ are studied in order to understand generalized symmetric spaces. The groups of type $E_6$ we consider here can be realized as both the group of linear maps that leave a certain determinant invariant, and also as the identity component of the automorphism group of a class of structurable algebras known as Brown algebras. We will classify the $k$-involutions of these groups of type $E_6$ using aspects of both descriptions, and give detailed descriptions of representatives over certain fields including algebraically closed fields, $\mathbb{R}$, $\mathbb{F}_p$, and $\mat…
Let $V$ and $W$ be finite dimensional real vector spaces and let $G\subset\GL(V)$ and $H\subset\GL(W)$ be finite subgroups. Assume for simplicity that the actions contain no reflections. Let $Y$ and $Z$ denote the real algebraic varieties corresponding to $\R[V]^G$ and $\R[W]^H$, respectively. If $V$ and $W$ are quasi-isomorphic, i.e., if there is a linear isomorphism $L\colon V\to W$ such that $L$ sends $G$-orbits to $H$-orbits and $L\inv$ sends $H$-orbits to $G$-orbits, then $L$ induces an isomorphism of $Y$ and $Z$. Conversely, suppose that $f\colon Y\to Z$ is a germ of a diffeomorphism sen…
Let $L$ be a Lie algebra with its enveloping algebra $U(L)$ over a field. In this paper we survey results concerning the isomorphism problem for enveloping algebras: given another Lie algebra $H$ for which $U(L)$ and $U(H)$ are isomorphic as associative algebras, can we deduce that $L$ and $H$ are isomorphic Lie algebras? Over a field of positive characteristic we consider a similar problem for restricted Lie algebras, that is, given restricted Lie algebras $L$ and $H$ for which their restricted enveloping algebras are isomorphic as algebras, can we deduce that $L$ and $H$ are isomorphic?
In this article we introduce an isomorphism invariant for type II_1 factors using the Connes-Folner condition. We compute bounds of this number for free group factors.
We prove that ``almost generically'' for a one-relator group Delzant's $T$-invariant (which measures the smallest size of a finite presentation for a group) is comparable in magnitude with the length of the defining relator. The proof relies on our previous results regarding isomorphism rigidity of generic one-relator groups and on the methods of the theory of Kolmogorov-Chaitin complexity. We also give a precise asymptotic estimate (when $k$ is fixed and $n$ goes to infinity) for the number $I_{k,n}$ of isomorphism classes of $k$-generator one-relator groups with a cyclically reduced defining…
Let $p$ be a an odd prime and let $G$ be a finite $p$-group with cyclic commutator subgroup $G'$. We prove that the exponent and the abelianization of the centralizer of $G'$ in $G$ are determined by the group algebra of $G$ over any field of characteristic $p$. If, additionally, $G$ is $2$-generated then almost all the numerical invariants determining $G$ up to isomorphism are determined by the same group algebras; as a consequence the isomorphism type of the centralizer of $G'$ is determined. These claims are known to be false for $p=2$.
Let $G$ be a complex reductive group and $V$ a $G$-module. Then the $m$th jet scheme $G_m$ acts on the $m$th jet scheme $V_m$ for all $m\geq 0$. We are interested in the invariant ring $\mathcal{O}(V_m)^{G_m}$ and whether the map $p_m^*\colon\mathcal{O}((V//G)_m) \rightarrow \mathcal{O}(V_m)^{G_m}$ induced by the categorical quotient map $p\colon V\rightarrow V//G$ is an isomorphism, surjective, or neither. Using Luna's slice theorem, we give criteria for $p_m^*$ to be an isomorphism for all $m$, and we prove this when $G=SL_n$, $GL_n$, $SO_n$, or $Sp_{2n}$ and $V$ is a sum of copies of the st…
