Zassenhaus conjecture for central extensions of $S_{5}$

Zassenhaus conjecture for $A_{6}$

For the alternating group $A_{6}$ of degree 6, Zassenhaus' conjecture about rational conjugacy of torsion units in integral group rings is confirmed.

Zassenhaus Conjecture on torsion units holds for $\operatorname{PSL}(2,p)$ with $p$ a Fermat or Mersenne prime

H.J. Zassenhaus conjectured that any unit of finite order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to an element of the form $\pm g$ with $g \in G$. Though known for some series of solvable groups, the conjecture has been proved only for thirteen non-abelian simple groups. We prove the Zassenhaus Conjecture for the groups $\operatorname{PSL}(2,p)$, where $p$ is a Fermat or Mersenne prime. This increases the list of non-abelian simple groups for which the conjecture is known by probably infinitely many, but at least …

On the first Zassenhaus conjecture and direct products

In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products as well as the General Bovdi Problem (Gen-BP) which turns out to be a slightly weaker variant of (ZC1). Among others we prove that (Gen-BP) holds for Sylow tower groups, so in particular for the class of supersolvable groups. (ZC1) is established for a direct product of Sylow-by-abelian groups provided the normal Sylow subgroups form together a Hall subgroup. We also show (ZC1) for certain direct products with one of the factors a Frobenius group. We extend the classical HeLP method to group ring…

Units in Blocks of Defect 1 and the Zassenhaus Conjecture

Building on previous work by Caicedo and the second author, we develop a method that decides the existence of units of finite order in blocks of $\mathbb{Z}_p G$ of defect 1. This allows us to prove that if $p$ is a prime and $G$ is a finite group whose Sylow $p$-subgroup has order $p$, then any unit $u\in \mathbb{Z} G$ of order $p$ is conjugate to an element of $\pm G$. This is a special case of the Zassenhaus conjecture. We also prove some new results on units of finite order in $\mathbb{Z} \mathrm{PSL}(2,q)$ for certain $q$, and construct a unit of order $15$ in $V(\mathbb{Z}_{(3,5)}\mathrm…

A Counterexample to the First Zassenhaus Conjecture

Hans J. Zassenhaus conjectured that for any unit $u$ of finite order in the integral group ring of a finite group $G$ there exists a unit $a$ in the rational group algebra of $G$ such that $a^{-1}\cdot u \cdot a=\pm g$ for some $g\in G$. We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order $2^7 \cdot 3^2 \cdot 5 \cdot 7^2 \cdot 19^2$ whose integral group ring contains a unit of order $7 \cdot 19$ which, in the rational group…

循环-阿贝尔群对第二和第三扎森豪斯猜想的反例

Cyclic-by-abelian counterexamples to the second and third Zassenhaus conjectures

该研究构造一类循环-阿贝尔有限群，得到无扎森豪斯分解的增广保持自同构，证明第二、第三扎森豪斯猜想对这类群不成立，解决了相关问题。

A Zassenhaus conjecture and CPA-structures on simple modular Lie algebras

Commutative post-Lie algebra structures on Lie algebras, in short CPA structures, have been studied over fields of characteristic zero, in particular for real and complex numbers motivated by geometry. A perfect Lie algebra in characteristic zero only admits the trivial CPA-structure. In this article we study these structures over fields of characteristic $p>0$. We show that every perfect modular Lie algebra in characteristic $p>2$ having a solvable outer derivation algebra admits only the trivial CPA-structure. This involves a conjecture by Hans Zassenhaus, saying that the outer derivation al…

关于单模李代数的Zassenhaus猜想的反例

Counterexamples to the Zassenhaus conjecture on simple modular Lie algebras

本文构造了Zassenhaus关于单模李代数外导代数可解性猜想的无限反例族，证明了特征3下特定维数的单模李代数 $H(2;(1,n))^{(2)}$ 的外导代数不可解，并给出了其外导代数的精确结构。

An algorithm to construct candidates to counterexamples to the Zassenhaus Conjecture

Let $G$ be a finite group, $N$ a nilpotent normal subgroup of $G$ and let $\mathrm{V}(\mathbb{\Z} G, N)$ denote the group formed by the units of the integral group ring $\mathbb{\Z} G$ of $G$ which map to the identity under the natural homomorphism $\mathbb{\Z} G \rightarrow \mathbb{\Z} (G/N)$. Sehgal asked whether any torsion element of $\mathrm{V}(\mathbb{\Z} G, N)$ is conjugate in the rational group algebra of $G$ to an element of $G$. This is a special case of the Zassenhaus Conjecture. By results of Cliff and Weiss and Hertweck, Sehgal's Problem has a positive solution if $N$ has at most …

Kimmerle conjecture for the Held and O'Nan sporadic simple groups

Using the Luthar--Passi method, we investigate the Zassenhaus and Kimmerle conjectures for normalized unit groups of integral group rings of the Held and O'Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held simple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture.

The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let $Z$ be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g* with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun …

Integral group ring of the McLaughlin simple group

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.

Integral group ring of the Mathieu simple group M24

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.

Integral group ring of the Suzuki sporadic simple group

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.

Torsion units in integral group ring of Higman-Sims simple group

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.

Integral group ring of Rudvalis simple 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.

Torsion units in integral group ring of the Mathieu simple group M22

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.

Integral group ring of the first Mathieu simple group

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.

From examples to methods: Two cases from the study of units in integral group rings

In this article, we review the proofs of the first Zassenhaus Conjecture on conjugacy of torsion units in integral group rings for the alternating groups of degree 5 and 6, by Luthar-Passi and Hertweck. We describe how the study of these examples led to the development of two methods -- the HeLP method and the lattice method. We exhibit these methods and summarize some results which were achieved using them. We then apply these methods to the study of the first Zassenhaus conjecture for the alternating group of degree 7 where only one critical case remains open for a full answer. Along the way…

Torsion units in integral group rings of Janko simple groups

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.

Rational conjugacy of torsion units in integral group rings of non-solvable groups

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…

Integral group ring of the Mathieu simple group M12

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.

Integral group ring of the Mathieu simple group M23

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.

Central extensions of current algebras

This is an old paper put here for archeological purposes. We compute the second cohomology of current Lie algebras of the form $L\otimes A$, where $L$ belongs to some class of Lie algebras which includes classical simple and Zassenhaus algebras, and of some modular semisimple Lie algebras. The results are largely superseded by subsequent papers, though, perhaps, some tricks and observations used here remain of minor interest.

On the torsion units of the integral group ring of finite projective special linear groups

H. J. Zassenhaus conjectured that any unit of finite order and augmentation one in the integral group ring of a finite group $G$ is conjugate in the rational group algebra to an element of $G$. One way to verify this is showing that such unit has the same distribution of partial augmentations as an element of $G$ and the HeLP Method provides a tool to do that in some cases. In this paper we use the HeLP Method to describe the partial augmentations of a hypothetical counterexample to the conjecture for the projective special linear groups.

A theorem of Hertweck on $p$-adic conjugacy of $p$-torsion units in group rings

A proof of a theorem of M. Hertweck presented during a seminar in January 2013 in Stuttgart is given. The proof is based on a preprint given to me by Hertweck. Let $R$ be a commutative ring, $G$ a finite group, $N$ a normal $p$-subgroup of $G$ and denote by $RG$ the group ring of $G$ over $R$. It is shown that a torsion unit $u$ in $\mathbb{Z}G$ mapping to the identity under the natural homomorphism $\mathbb{Z}G \rightarrow \mathbb{Z}G/N$ is conjugate in the unit group of $\mathbb{Z}_pG$ to an element in $N$. Here $\mathbb{Z}_p$ denotes the $p$-adic integers. The result is achieved proving a r…

The Twisted Derivation Problem for Group Rings

We study $(σ,τ)$-derivations of a group ring $RG$ where $G$ is a group with center having finite index in $G$ and $R$ is a semiprime ring with $1$ such that either $R$ has no torsion elements or that if $R$ has $p$-torsion elements, then $p$ does not divide the order of $G$ and let $σ,τ$ be $R$-linear endomorphisms of $RG$ fixing the center of $RG$ pointwise. We generalize Main Theorem $1.1$ of \cite{Chau-19} and prove that there is a ring $T\supset R$ such that $\mathcal{Z}(T)\supset\mathcal{Z}(R)$ and that for the natural extensions of $σ, τ$ to $TG$ we get $H^1(TG,{}_σTG_τ)=0$, where ${}_σT…