Leo Margolis

Zassenhaus Conjecture for cyclic-by-abelian groups

Zassenhaus Conjecture for torsion units states that every aug- mentation one torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of rational group algebra QG. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. We prove the conjecture for cyclic-by-abelian groups. In this paper G is a finite group and RG denotes the group ring of G with coefficients in a ringR. The units of RG of augmentation one are usually called normalized units. In the 1960s Hans Zassenhaus established a series of conjectures about the finite subgroups of normalized units of ZG. Namely he conjectured that every finite group of normalized units of ZG is conjugate to a subgroup of G in the units of QG. These conjecture is usually denoted (ZC3), while the version of (ZC3) for the particular case of subgroups of normalized units with the same cardinality as G is usually denoted (ZC2). These conjectures have important consequences. For example, a positive solution of (ZC2) implies a positive solution for the Isomorphism and Automorphism Problems (see (Seh93) for details). The most celebrated positive result for Zassenhaus Conjectures is due to Weiss (Wei91) who proved (ZC3) for nilpotent groups. However Roggenkamp and Scott founded a counterexample to the Automorphism Problem, and henceforth to (ZC2) (see (Rog91) and (Kli91)). Later Hertweck (Her01) provided a counterexample to the Isomorphism Problem. The only conjecture of Zassenhaus that is still up is the version for cyclic sub- groups namely:

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 PSL(2, 19). We also prove the Zassenhaus conjecture for 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 PGL(2, 9). This completes the proof of a theorem of Kimmerle and Konovalov that shows that the prime graph question has an affirmative answer for all groups having an order divisible by at most three different primes.

HeLP -- A GAP-package for torsion units in integral group rings

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.

On the Prime Graph Question for Integral Group Rings of 4-Primary Groups II

We study the Prime Graph Question for integral group rings. This question can be reduced to almost simple groups by a result of Kimmerle and Konovalov. We prove that the Prime Graph Question has an affirmative answer for all almost simple groups having a socle isomorphic to PSL(2,pf) for f ≤ 2, establishing the Prime Graph Question for all groups where the only non-abelian composition factors are of the aforementioned form. Using this, we determine exactly how far the so-called HeLP method can take us for (almost simple) groups having an order divisible by at most four different primes.In this article the study of the Prime Graph Question for the integral group ring of almost simple groups which have an order divisible by exactly 4 different primes is continued. We provide more details on the recently developed “lattice method” which involves the calculation of Littlewood-Richardson coefficients. We apply the method obtaining results complementary to those previously obtained using the HeLP-method. In particular the “lattice method” is applied to infinite series of groups for the first time. We also prove the Zassenhaus Conjecture for four more simple groups. Furthermore we show that the Prime Graph Question has a positive answer around the vertex 3 provided the Sylow 3-subgroup is of order 3.

On the prime graph question for integral group rings of 4-primary groups I

We study the Prime Graph Question for integral group rings. This question can be reduced to almost simple groups by a result of Kimmerle and Konovalov. We prove that the Prime Graph Question has an affirmative answer for all almost simple groups having a socle isomorphic to

The status of the Zassenhaus conjecture for small groups

\operatorname{PSL}(2, p^f)

Algorithmic Aspects of Units in Group Rings

for

Subgroup Isomorphism Problem for units of integral group rings

f \leq 2

The Herzog–Schönheim conjecture for small groups and harmonic subgroups

, establishing the Prime Graph Question for the first time for all automorphic extensions of series of simple groups. Using this, we determine exactly how far the so-called HeLP-method can take us for (almost simple) groups having an order divisible by at most

A Sylow theorem for the integral group ring of PSL(2,q)

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