V. A. Bovdi, A. B. Konovalov
We investigated the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the simple Mathieu group M11. As a consequence, for this group we confirm the conjecture by Kimmerle about prime graphs.
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We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group $M_{24}$. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.
We investigate the possible character values of torsion units of the normalized unit group of the integral group ring of Mathieu sporadic group $M_{22}$. We confirm the Kimmerle conjecture on prime graphs for this group and specify the partial augmentations for possible counterexamples to the stronger Zassenhaus conjecture.
We investigate the classical Zassenhaus conjecture for the unit group of the integral group ring of Mathieu simple group M23 using the Luthar-Passi method. This work is a continuation of the research that we carried out for Mathieu groups M11 and M12. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.
We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the Mathieu sporadic group M12. As a consequence, we confirm for this group the Kimmerle's conjecture on prime graphs.
We describe all Mathieu-Zhao spaces of $k[x_1,\cdots,x_n]$ ($k$ is an algebraically closed field of characteristic zero) which contains an ideal of finite codimension. Furthermore we give an algorithm to decide if a subspace of the form $I+kv_1+\cdots+kv_r$ is a Mathieu-Zhao space, in case the ideal $I$ has finite codimension.
Let $G$ be a group, $R$ an integral domain, and $V_G$ the subspace of the group algebra $R[G]$ consisting of all the elements of $R[G]$ whose coefficient of the identity element $1_G$ of $G$ is equal to zero. Motivated by the Mathieu conjecture [M], the Duistermaat-van der Kallen theorem [DK], and also by recent studies on the notion of Mathieu subspaces introduced in [Z4] and [Z6], we show that for finite groups $G$, $V_G$ under certain conditions also forms a Mathieu subspace of the group algebra $R[G]$. We also show that for the free abelian groups $G=\Bbb Z^n$ $(n\ge 1)$ and any integral d…
This is the note for the four lectures given by the author in the ``International Short-School/Conference on Affine Algebraic Geometry and the Jacobian Conjecture" at Chern Institute of Mathematics, Nankai University, Tianjin, China. July 14-25, 2014. The aim of this lectures is to give an introduction to the theory of Mathieu subspaces. We will not treat all topics in their most general setting, but will restrict to certain classes of "nice" rings. As we shall see even for these rings there is a lot we don't know yet!
A Mathieu-Zhao subspace is a generalization of an ideal of an associative algebra $\mathcal A$ over a unital ring $R$ first formalized in 2010. A vertex algebra is an algebraic structure first developed in conjunction with string theory in the 1960s and later axiomatized by mathematicians in the 1980s. We formally introduce the definition of a Mathieu-Zhao subspace $M$ of a vertex algebra $V$. Building on natural connections to associative algebras, we classify an infinite set of non-trivial, non-ideal Mathieu-Zhao subspaces for simple and general vertex algebras by group action eigenspace dec…
Motivated by the Mathieu conjecture [Ma], the image conjecture [Z3] and the well-known Jacobian conjecture [K] (see also [BCW] and [E]), the notion of Mathieu subspaces as a natural generalization of the notion of ideals has been introduced recently in [Z4] for associative algebras. In this paper, we first study algebraic elements in the radicals of Mathieu subspaces of associative algebras over fields and prove some properties and characterizations of Mathieu subspaces with algebraic radicals. We then give some characterizations or classifications for strongly simple algebras (the algebras wi…
The aim of this article is to draw attention towards various natural but unanswered questions related to the lower central series of the unit group of an integral group ring.
We consider the Zassenhaus conjecture for the normalized unit group of the integral group ring of the McLaughlin sporadic group McL. As a consequence, we confirm for this group the Kimmerle's conjecture on prime graphs.
Using the Luthar--Passi method, we investigate the possible orders and partial augmentations of torsion units of the normalized unit group of integral group rings of Conway simple groups $Co_1$, $Co_2$ and $Co_3$.
In this paper, we study the question of when the symmetric units in an integral group ring ZG form a multiplicative group. When G is periodic, necessary and sufficient conditions are given for this to occur.
Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of integral group rings of Janko sporadic simple groups. As a consequence, we obtain that the Gruenberg-Kegel graph of the Janko groups $J_1$, $J_2$ and $J_3$ is the same as that of the normalized unit group of their respective integral group ring.
Using the Luthar--Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Suzuki sporadic simple group Suz. As a consequence, for this group we confirm the Kimmerle's conjecture on prime graphs.
Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Higman-Sims simple sporadic group HS. As a consequence, we confirm the Kimmerle's conjecture on prime graphs for this sporadic group.
Using the Luthar-Passi method, we investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the Rudvalis sporadic simple group Ru. As a consequence, for this group we confirm Kimmerle's conjecture on prime graphs.
We classify the finite groups $G$ such that the group of units of the integral group ring ${\mathbb Z} G$ has a subgroup of finite index which is a direct product of free-by-free groups.
We give a list of finite groups containing all finite groups $G$ such that the group of units $\Z G^*$ of the integral group ring $\Z G$ is subgroup separable. There are only two types of these groups $G$ for which we cannot decide wether $ZG^*$ is subgroup separable, namely the central product $Q_8 Y D_8$ and $Q_8\times C_p{with} p \text{prime and} p\equiv -1 \mod (8)$.
For G = PSL(2,p^f) denote by ZG the integral group ring, by V(ZG) the group of normalized units of ZG and let r be a prime different from p. Using the so called HeLP-method we prove, that units of r-power order in V(ZG) are rationally conjugate to elements of G. As a consequence we prove, that subgroups of prime power order in V(ZG) are rationally conjugate to subgroups of G, provided p = 2 or f =1.
Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is isomorphic to a subgroup of $G$. Furthermore, it is shown that a composition factor of a finite subgroup of $V(\mathbb{Z}G)$ is isomorphic to a subgroup of $G$.
The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $\mathbb Z G$ , i.e. the prime graph of the normalised unit group of $\mathbb Z G$ coincides with that one of the group $G$. In this note we prove for finite groups $G$ a reduction of the prime graph question to almost simple groups. We apply this reduction to finite groups $G$ whose order is divisible by at most three primes and show that the Gruenberg - Kegel graph of such groups coincides with the prime graph of $G$.
It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring of a finite group is conjugate to a group element within the rational group algebra. The object of this note is the computational aspect of a method developed by I.S. Luthar and I.B.S. Passi which sometimes permits an answer to this conjecture. We illustrate the method on certain explicit examples. We prove with additional arguments that the conjecture is valid for any 3-dimensional crystallographic point group. Finally we apply the method to generic character tables and establish a p-variation of the conjecture …
Inner relations are derived between partial augmentations of certain elements (units or idempotents) in group rings.
The Prime Graph Question for integral group rings asks if it is true that if the normalized unit group of the integral group ring of a finite group $G$ contains an element of order $pq$, for some primes $p$ and $q$, also $G$ contains an element of that order. We answer this question for the three Conway sporadic simple groups after reducing it to a combinatorial question about Young tableaus and Littlewood-Richardson coefficients. This finishes work of V. Bovdi, A. Konovalov and S. Linton.
The object of this paper is to examine finite solvable groups whose integral group rings have only trivial central units.
We show that for every prime $r$ all $r$-subgroups in the normalized units of the integral group ring of $\operatorname{PSL}(2,p^3)$ are isomorphic to subgroups of $\operatorname{PSL}(2,p^3)$. This answers a question of M. Hertweck, C.R. Höfert and W. Kimmerle for this series of groups.
The object of study in this paper is the finite groups whose integral group rings have only trivial central units. Prime-power groups and metacyclic groups with this property are characterized. Metacyclic groups are classified according to the central height of the unit groups of their integral group rings.
We introduce the notion of the Frobenius-Perron dimension of an integral $\Bbb Z_+$-ring and give some applications of this notion to classification of finite dimensional quasi-Hopf algebras with a unique nontrivial simple module, and of quasi-Hopf and Hopf algebras of prime dimension $p$.
We prove that every non-finitely generated projective module over the integral group ring of a polycyclic-by-finite group G is free if and only if G is polycyclic.
Let $R$ be a commutative ring and $\mathcal A$ an $R$-algebra. An $R$-$\mathcal E$-derivation of $\mathcal A$ is an $R$-linear map of the form $\operatorname{I}-ϕ$ for some $R$-algebra endomorphism $ϕ$ of $\mathcal A$, where $\operatorname{I}$ denotes the identity map of $\mathcal A$. In this paper we discuss some open problems on whether or not the image of a locally finite $R$-derivation or $R$-$\mathcal E$-derivation of $\mathcal A$ is a Mathieu subspace [Z2, Z3] of $\mathcal A$, and whether or not a locally nilpotent $R$-derivation or $R$-$\mathcal E$-derivation of $\mathcal A$ maps every …
The augmentation powers in an integral group ring $\mathbb ZG$ induce a natural filtration of the unit group of $\mathbb ZG$ analogous to the filtration of the group $G$ given by its dimension series $\{D_n(G)\}_{n\ge 1}$. The purpose of the present article is to investigate this filtration, in particular, the triviality of its intersection.
In calculus, an indefinite integral of a function $f$ is a differentiable function $F$ whose derivative is equal to $f$. In present paper, we generalize this notion of the indefinite integral from the ring of real functions to any ring. The main goal of the paper is to focus on the properties of such generalized integrals that are inherited from the well-known basic properties of indefinite integrals of real functions.
In this paper, we exhibit the creation of the maximal integral domain mid(R) generated by a nonzero commutative ring R.
We prove that the Zassenhaus conjecture is true for $PSL(2,8)$ and $PSL(2,17)$. This is a continuation of research initiated by W. Kimmerle, M. Hertweck and C. Höfert.
The Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups U it is true that if U is isomorphic to a subgroup of V(ZG), the group of normalized units of the integral group ring of the finite group G, it must be isomorphic to a subgroup of G. The smallest groups known not to satisfy this property are the counterexamples to the Isomorphism Problem constructed by M. Hertweck. However the only groups known to satisfy it are cyclic groups of prime power order and elementary-abelian p-groups of rank 2. We prove the Subgroup Isomorphism Problem for C_4 x C_2. Moreover we p…
It is shown that for any torsion unit of augmentation one in the integral group ring $\mathbb{Z} G$ of a finite solvable group $G$, there is an element of $G$ of the same order.
Classifying integral Grothendieck rings up to rank 5 and beyond
本文定义了可范畴化的整Grothendieck环，利用Normaliz工具、Frobenius-Schur指标和Galois理论，完成了秩至5及部分高秩情况的完整分类，并发现了首个非Isaacs整fusion category等新结构。
Eventually periodic resolutions with applications to integral group rings
本文提出一种利用 Shamash 构造和迭代映射锥技术，系统清除复形同调以构造最终周期投射分解的方法，并应用于双曲三角群等群的整群环计算。
Finite groups in integral group rings
本文回顾了用于多场合教学与会议的笔记内容，聚焦有限群整群环单位群的有限子群相关问题及解决技术。
Let $p$ be a prime such that the multiplicative order $m$ of $2$ modulo $p$ is even. We prove that the integral group ring $\mathbb{Z}[Q_8 \times C_p]$ has the multiplicative Jordan decomposition property when $m$ is congruent to $2$ modulo $4$. There are infinitely many such primes and these primes include the case $p \equiv 3 \pmod{4}$. We also prove that $\mathbb{Z}[Q_8 \times C_5]$ has the multiplicative Jordan decomposition property in a new way.
During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group $\U (\Z G)$ of the integral group ring $\Z G$ of a finite group $G$. These constructions rely on explicit constructions of units in $\Z G$ and proofs of main results make use of the description of the Wedderburn components of the rational group algebra $\Q G$. The latter relies on explicit constructions of primitive central idempotents and the rational representations of $G$. It turns out that the existence of reduced two degree repres…
In this paper we introduce new modules over the ring of ponderation functions, so we recover old results in harmonic analysis from the side of ring theory. Moreover, we prove that Laplace transform, Fourier transform and Hankel transform generate some kind of modules over the ring of ponderation functions.
Integral Dependence over One-Sided Ideals, Part II: G-Left Ideals and the Köthe Problem
本文通过G-左理想研究左理想整相关，刻画其交集为上幂零根，并引入左KI-环类，证明左Artin环、PI-环、代数代数及小维数代数为左KI-环，关联Köthe猜想。
Rigid integral representations of quivers over arbitrary commutative rings
本文将作者早期关于主理想环上由有限生成自由模构成的箭图刚性表示分类结果，推广至任意交换环上由有限生成投射模构成的箭图表示。
Let R be a Lie nilpotent algebra of index k over a field K of characteristic zero. If G is an n-element subgroup of Aut(R) of the K-automorphisms, then we prove that R is right integral over Fix(G) of degree n^k. In the presence of a primitive n-th root of unity e in K, for a K-automorphism d in Aut(R) with d^n=id, we prove that the skew polynomial algebra R[w,d] is right integral of degree n^k over Fix(d)[w^n].
We describe an algorithm for obtaining generators of the unit group of the integral group ring ZG of a finite abelian group G. We used our implementation in Magma of this algorithm to compute the unit groups of ZG for G of order up to 110. In particular for those cases we obtained the index of the group of Hoechsmann units in the full unit group. At the end of the paper we describe an algorithm for the more general problem of finding generators of an arithmetic group corresponding to a diagonalizable algebraic group.
Let $D$ be an integral domain with quotient field $K$ and $Ω$ a finite subset of $D$. McQuillan proved that the ring ${\rm Int}(Ω,D)$ of polynomials in $K[X]$ which are integer-valued over $Ω$, that is, $f\in K[X]$ such that $f(Ω)\subset D$, is a Prüfer domain if and only if $D$ is Prüfer. Under the further assumption that $D$ is integrally closed, we generalize his result by considering a finite set $S$ of a $D$-algebra $A$ which is finitely generated and torsion-free as a $D$-module, and the ring ${\rm Int}_K(S,A)$ of integer-valued polynomials over $S$, that is, polynomials over $K$ whose i…
Let $\mathcal{Z}(\mathcal{U}(\mathbb{Z}[G]))$ denote the group of central units in the integral group ring $\mathbb{Z}[G]$ of a finite group $G$. A bound on the index of the subgroup generated by a virtual basis in $\mathcal{Z}(\mathcal{U}(\mathbb{Z}[G]))$ is computed for a class of strongly monomial groups. The result is illustrated with application to the groups of order $p^{n}$, $p$ prime, $n \leq 4$. The rank of $\mathcal{Z}(\mathcal{U}(\mathbb{Z}[G]))$ and the Wedderburn decomposition of the rational group algebra of these $p$-groups have also been obtained.
A finite group $G$ is said to have the nilpotent decomposition property (ND) if for every nilpotent element $α$ of the integral group ring $\mathbb{Z}[G]$ one has that $αe$ also belong to $\mathbb{Z}[G]$, for every primitive central idempotent $e$ of the rational group algebra $\mathbb{Q}[G]$. Results of Hales, Passi and Wilson, Liu and Passman show that this property is fundamental in the investigations of the multiplicative Jordan decomposition of integral group rings. If $G$ and all its subgroups have ND then Liu and Passman showed that $G$ has property SSN, that is, for subgroups $H$, $Y$ …
It is shown that in the units of augmentation one of an integral group ring $\mathbb{Z} G$ of a finite group $G$, a noncyclic subgroup of order $p^{2}$, for some odd prime $p$, exists only if such a subgroup exists in $G$. The corresponding statement for $p=2$ holds by the Brauer--Suzuki theorem, as recently observed by W. Kimmerle.
Let $D$ be a domain with fraction field $K$, and let $M_n(D)$ be the ring of $n \times n$ matrices with entries in $D$. The ring of integer-valued polynomials on the matrix ring $M_n(D)$, denoted ${\rm Int}_K(M_n(D))$, consists of those polynomials in $K[x]$ that map matrices in $M_n(D)$ back to $M_n(D)$ under evaluation. It has been known for some time that ${\rm Int}_{\mathbb{Q}}(M_n(\mathbb{Z}))$ is not integrally closed. However, it was only recently that an example of a polynomial in the integral closure of ${\rm Int}_{\mathbb{Q}}(M_n(\mathbb{Z}))$ but not in the ring itself appeared in t…
Zeta functions for table algebras and fusion rings with irrational-valued characters
本文针对具有无理值不可约特征标的秩3标准整表代数与整融合环，通过显式计算zeta积分，求出了相应序的zeta函数。
For a finite group $G$, let $\tilde{\mathbb{Z}}$ be the semilocalization of $\mathbb{Z}$ at the prime divisors of $|G|$. If $G$ is a Frobenius group with Frobenius kernel $K$, it is shown that each torsion unit in the group ring $\tilde{\mathbb{Z}} G$ which maps to the identity under the natural ring homomorphism $\tilde{\mathbb{Z}} G \rightarrow \tilde{\mathbb{Z}} G/K$ is conjugate to an element of $G$ by a unit in $\tilde{\mathbb{Z}} G$.
We introduce a new method to study rational conjugacy of torsion units in integral group rings using integral and modular representation theory. Employing this new method, we verify the first Zassenhaus Conjecture for the group $\operatorname{PSL}(2,19)$. We also prove the Zassenhaus Conjecture for $\operatorname{PSL}(2,23)$. In a second application we show that there are no normalized units of order $6$ in the integral group rings of $M_{10}$ and $\operatorname{PGL}(2,9)$. This completes the proof of a theorem of W. Kimmerle and A. Konovalov that the Prime Graph Question has an affirmative an…
Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on algebras. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true for Hopf algebras over rings. We show that over any commutative ring R that is not a field there exists a Hopf algebra H over R containing a non-zero integral but not being finitely generated as R-module. On the contrary we show that Sweedler's equivalence is still valid fo…
We study integrality over rings (all commutative in this paper) and over ideal semifiltrations (a generalization of integrality over ideals). We begin by reproving classical results, such as a version of the "faithful module" criterion for integrality over a ring, the transitivity of integrality, and the theorem that sums and products of integral elements are again integral. Then, we define the notion of integrality over an ideal semifiltration (a sequence $\left( I_0,I_1,I_2,\ldots\right)$ of ideals satisfying $I_0 =A$ and $I_a I_b \subseteq I_{a+b}$ for all $a,b\in\mathbb{N}$), which general…
Given an integer $n$, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in $\{1,2,\dots, n-1\}$. Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is definitely periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular …
It is shown that every dp-minimal integral domain $R$ is a local ring and for every non-maximal prime ideal $\mathfrak p $ of $R$, the localization $R_{\mathfrak p }$ is a valuation ring and $\mathfrak{p}R_{\mathfrak{p}}=\mathfrak{p}$. Furthermore, a dp-minimal integral domain is a valuation ring if and only if its residue field is infinite or its residue field is finite and its maximal ideal is principal.
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