V. A. Bovdi, E. Jespers, A. B. Konovalov
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.
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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 $Co_3$.
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.
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.
The connections between the properties of associative rings that are Lie-solvable (Engel, n-Engel, locally finite, respectively) and the properties of their adjoin subgroups are investigated.
The torsion ideal lattice of the endomorphism ring of an abelian p-group
本文通过基数不变量系统分类了阿贝尔p-群自同态环的挠理想的格。
We use a recent result of Alexander and Nishinaka to show that if $G$ is a non-elementary torsion-free hyperbolic group and $R$ is a countable domain, then the group ring $RG$ is primitive. This implies that the group ring $KG$ of any non-elementary torsion-free hyperbolic group $G$ over a field $K$ is primitive.
Nontrivial pairs of zero-divisors in group rings are introduced and discussed. A problem on the existence of nontrivial pairs of zero-divisors in group rings of free Burnside groups of odd exponent $n \gg 1$ is solved in the affirmative. Nontrivial pairs of zero-divisors are also found in group rings of free products of groups with torsion.
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.
We briefly summarize the background of the HeLP-method for torsion units in group rings and present some functionality of a GAP-package implementing it.
We use the theory of blocks of cyclic defect to prove that under a certain condition on the principal p-block of a finite group G the normalized unit group of the integral group ring of G contains an element of order pq if and only if so does G, for q a prime different from p. Using this we verify the Prime Graph Question for all alternating and symmetric groups and also for two sporadic simple groups.
Let $Γ$ be a discrete group. Following Linnell and Schick one can define a continuous ring $c(Γ)$ associated with $Γ$. They proved that if the Atiyah Conjecture holds for a torsion-free group $Γ$, then $c(Γ)$ is a skew field. Also, if $Γ$ has torsion and the Strong Atiyah Conjecture holds for $Γ$, then $c(Γ)$ is a matrix ring over a skew field. The simplest example when the Strong Atiyah Conjecture fails is the lamplighter group $Γ=Z_2\wr Z$. It is known that $C(Z_2\wr Z)$ does not even have a classical ring of quotients. Our main result is that if $H$ is amenable, then $c(Z_2\wr H)$ is isomor…
Kaplanski's Zero Divisor Conjecture envisions that for a torsion-free group G and an integral domain R, the group ring R[G] does not contain non-trivial zero divisors. We define the length of an element a in R[G] as the minimal non-negative integer k for which there are ring elements r_1,...,r_k in R and group elements g_1,...,g_k in G such that a = r_1 g_1+...+r_k g_k. We investigate the conjecture when R is the field of rational numbers. By a reduction to the finite field with two elements, we show that if ab = 0 for non-trivial elements in the group ring of a torsion-free group over the rat…
We describe the main questions connected to torsion subgroups in the unit group of integral group rings of finite groups and algorithmic methods to attack these questions. We then prove the Zassenhaus Conjecture for Amitsur groups and prove that any normalized torsion subgroup in the unit group of an integral group of a Frobenius complement is isomorphic to a subgroup of the group base. Moreover we study the orders of torsion units in integral group rings of finite almost quasisimple groups and the existence of torsion-free normal subgroups of finite index in the unit group.
We show that the units found in torsion-free group rings by Gardam are twisted unitary elements. This justifies some choices in Gardam's construction that might have appeared arbitrary, and yields more examples of units. We note that all units found up to date exhibit non-trivial symmetry.
Generalized torsion in triangular matrix groups
研究交换环上上三角矩阵群中广义挠元的刻画，并基于环的单位群的挠与指数性质确定这些群何时满足广义恒等式，构造了满足指定长度广义恒等式的无限有限生成可解群族，并给出Sherman问题的反例。
Division rings for group algebras of virtually compact special groups and $3$-manifold groups
本文为无挠虚拟紧特殊群和有限生成无挠3维流形群的群代数构造除环嵌入，证明其在局部可指示时为Hughes自由，并推出零因子猜想、Kielak–Linton猜想及凝聚性等结论。
For any odd prime $p$ we consider representations of a group of order $p$ in the symplectic group $Sp(p-1,Z[1/n])$ of $(p-1)\times(p-1)$-matrices over the ring $Z[1/n]$, $0\neq n\in N$. We construct a relation between the conjugacy classes of subgroups $P$ of order $p$ in the symplectic group and the ideal class group in the ring $Z[1/n]$. This is used for the study of these classes. In particular we determine the centralizer $C(P)$ and $N(P)/C(P)$ where $N(P)$ denotes the normalizer.
Let R be any ring (with 1), Γa group and RΓthe corresponding group ring. Let Ext_{RΓ}^{*}(M,M) be the cohomology ring associated to the RΓ-module M. Let H be a subgroup of finite index of Γ. The following is a special version of our main Theorem: Assume the profinite completion of Γis torsion free. Then an element ζin Ext_{RΓ}^{*}(M,M) is nilpotent (under Yoneda's product) if and only if its restriction to Ext_{RH}^{*}(M,M)$ is nilpotent. In particular this holds for the Thompson group. There are torsion free groups for which the analogous statement is false.
In the present notes we generalize the classical work of Demazure [Invariants symétriques entiers des groupes de Weyl et torsion] to arbitrary oriented cohomology theories and formal group laws. Let G be a split semisemiple linear algebraic group over a field and let T be its split maximal torus. We construct a generalized characteristic map relating the so called formal group ring of the character group of T with the cohomology of the variety of Borel subgroups of G. The main result of the paper says that the kernel of this map is generated by W-invariant elements, where W is the Weyl group o…
We give a survey of the theory of surface braid groups and the lower algebraic K-theory of their group rings. We recall several definitions and describe various properties of surface braid groups, such as the existence of torsion, orderability, linearity, and their relation both with mapping class groups and with the homotopy groups of the 2-sphere. The braid groups of the 2-sphere and the real projective plane are of particular interest because they possess elements of finite order, and we discuss in detail their torsion and the classification of their finite and virtually cyclic subgroups. F…
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