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Martin Hertweck

Publications and source records attributed to Martin Hertweck.

10 recordsLinked to original sources
arXiv2012

On torsion units in integral group rings of Frobenius groups

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 $\tilde{\mathbb{Z}} G$.

math.RT↗
arXiv2008

Finite groups of units and their composition factors in the integral group rings of the groups $\text{PSL}(2,q)$

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 $G$.

math.GR↗
arXiv2008

Class-preserving automorphisms and the normalizer property for Blackburn groups

For a group $G$, let $U$ be the group of units of the integral group ring $\mathbb{Z}G$. The group $G$ is said to have the normalizer property if $\text{N}_U(G)=\text{Z}(U)G$. It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being nontrivial. Groups $G$ for which class-preserving automorphisms are inner automorphisms, $\text{Out}_c(G)=1$, have the normalizer property. Recently, Herman and Li have shown that $\text{Out}_c(G)=1$ for a finite Blackburn group $G$. We show that $\text{out}_c(G)=1$ for the members $G$ of a few classes of metabelian groups, from which the Herman--Li result follows. Together with recent work of Hertweck, Iwaki, Jespers and Juriaans, our main result implies that, for an arbitrary group $G$, the group of hypercentral units of $U$ is contained in $\text{Z}(U)G$.

math.GR↗
arXiv2007

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.

math.RT↗
arXiv2007

Partial augmentations and Brauer character values of torsion units in group rings

For a torsion unit $u$ of the integral group ring $\mathbb{Z} G$ of a finite group $G$, and a prime $p$ which does not divide the order of $u$ (but the order of $G$), a relation between the partial augmentations of $u$ on the $p$-regular classes of $G$ and Brauer character values is noted, analogous to the obvious relation between partial augmentations and ordinary character values. For non-solvable $G$, consequences concerning rational conjugacy of $u$ to a group element are discussed, considering as examples the symmetric group $S_{5}$ and the groups $\text{\rm PSL}(2,p^{f})$.

math.RA↗
arXiv2006

Units of $p$-power order in principal $p$-blocks of $p$-constrained groups

Let $G$ be a finite group having a normal $p$-subgroup $N$ that contains its centralizer $\text{C}_{G}(N)$, and let $R$ be a $p$-adic ring. It is shown that any finite $p$-group of units of augmentation one in $RG$ which normalizes $N$ is conjugate to a subgroup of $G$ by a unit of $RG$, and if it centralizes $N$ it is even contained in $N$.

math.RT↗
arXiv2006

Isomorphisms between centers of integral group rings

For finite nilpotent groups $G$ and $G^{\prime}$, and a $G$-adapted ring $S$ (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings $SG$ and $SG^{\prime}$ is monomial, i.e., maps class sums in $SG$ to class sums in $SG^{\prime}$ up to multiplication with roots of unity. As a consequence, $G$ and $G^{\prime}$ have identical character tables if and only if the centers of their integral group rings $\mathbb{Z} G$ and $\mathbb{Z} G^{\prime}$ are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.

math.RT↗