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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - ZC (Torsion units of integral group rings)

Teaser

Summary

The problem studied in this project was the so-called Zassenhaus Conjecture. Hans Julius Zassenhaus (1912-1991) made important contributions to algebra. Group rings were introduced in the late 19th century as a tool to study symmetries, matrices and other algebraic objects. The group ring RG of a group G over a ring R can be defined as a vector space over R where the base is given by the elements of the group G. This provides an additive structure and the multiplicative structure on RG is given by extending the multiplications on G and R and declaring elements of G and R to commute.

From the middle of the 20th century the group ring became an object of interest in itself. It can be seen as a structure joining in an elegant manner the algebraic theories on rings and group. When R=Z, the ring of integers number theory enters the picture and makes the closest connection to the group base G, since it keeps the arithmetic information which would be lost when one is allowed to divide by some primes. The first to study particularly the unit group of the integral group ring was Graham Higman in his PhD thesis in 1940 who proved that for abelian G the units of finite order in ZG are, up to sign, exactly the elements of the group base G. This gave support to a conjecture that was probably mentioned by the specialist during that years and that was publised formally by Zassenhaus in 1974. This conjecture stated that if G is a finite group then a unit of finite order in ZG should be should be conjugate in the rational group algebra, up to sign, to an element of the group base. This conjecture inspired many research carried out during the following decade and it became one of the central problems in the study of integral group rings.

The impact of mathematics on society is manifold and often comes at a later stage in a very surprising manner. For example the ideal of groups rings now play a role in cryptography, but were originally studied as an object of pure scientific interest. It is hence impossible to predict, as mostly with fundamental research, which could be the consequences and practically useful implementations of this research.

Concretely the objectives of this project were to study the Zassenhaus Conjeture for several new classes of groups, which were more concretely the class of metabelian groups and the projective special linear groups PSL(2,q). Also the project included the study of a weaker version of the Zassenhaus Conjeture, the so-called Prime Graph Question, introduced by Wolfgang Kimmerle in 2006.

Work performed

We were able to achieve far reaching results regarding the Zassehaus Conjecture and in the end even to give a general negative answer to the conjecture by providing a specific counterexample to it.

In a project in collaboration with Andreas BÃ¤chle (VUB, Belgium), Allen Herman and Gurmail Singh (University of Regina, Canada) and Alexander Konovalov (University of St Andrews, UK) we proved the conjecture for groups of order at most 143. We studied the conjecture for metabelian groups using ideas inspired by research of Cliff and Weiss and Hertweck to obtain the exact limits of what was possible using our methods. It then turned out that these limits correspond quite closely to the limits of the validity of the conjecture in this class of groups, as found in the research constituting the counterexample for the Zassenhaus Conjecture in collaboration with Florian Eisele (City University, London, UK).

Regarding non-solvable groups we proved the conjecture for the projective special linear groups PSL(2,p), if p is a Fermat or Mersenne prime. This was done in collaboration with the local PhD student Mariano Serrano who then continued to prove the conjecture for the special linear groups SL(2,p), for any prime p, thus obtaining more mathematical independence.

Regarding the Prime Graph Question it was studied for alternating and symmetric groups in collaboration with Andreas BÃ¤chle and Mauricio Caicedo (VUB, Belgium) and Eugenio Giannelli (Technical University, Kaiserslautern, Germany/ University of Cambridge, UK). Moreover it was studied for sporadic simple groups.

In collaboration with Ofir Schnabel (University of Stuttgart, Germany) twisted group rings were investigated and a new line of research was started.

We provide a full list of publications related to the project below. Most of them are still in the process of being peer-reviewed, but they appeared already in an OpenSource repository.

The results of this project were also at various stages presented to specialized audiences at conferences and in seminars. The experienced researcher gave 8 talks, the supervisor and his PhD student 6 talks and other collaborators of the project 7 talks. The project was presented to the general public at two occasions during the European Research Night in Murcia.

- A. BÃ¤chle, A. Herman, A. Konovalov, L. Margolis, and G. Singh, The status of the
Zassenhaus conjecture for small groups, Experimental Mathematics (2017), 6 pages,
doi:10.1080/10586458.2017.1306814.

- A. BÃ¤chle, W. Kimmerle, and L. Margolis, Algorithmic aspects of units in group
rings, to be published in a proceedings volume of the DFG priorety program 1489,
arxiv.org/abs/1612.06171 (2016), 21 pages.

- A. BÃ¤chle and L. Margolis, On the Prime Graph Question for Integral Group Rings of
4-primary groups II, preprint, arxiv.org/abs/1606.01506 (2016), 17 pages.

- A. BÃ¤chle and L. Margolis, On the prime graph question for integral group rings
of 4-primary groups I, Internat. J. Algebra Comput. 27 (2017), no. 6, 731â€“767.

- F. Eisele and L Margolis, A counterexample to the first zassenhaus conjecture,
http://arxiv.org/abs/1710.08780 (2017), 32 pages.

- L. Margolis, A Theorem of Hertweck on p-adic conjugacy,
http://arxiv.org/abs/1706.02117 (2017), 11 pages.

- L. Margolis and Ã. del RÃo, An algorithm to construct candidates to counterexamples to the Zassenhaus Conjecture
of integral group rings, preprint, arxiv.org/abs/1710.05629 (2017), 21 pages.

- L. Margolis and Ã. del RÃo, Cliff-Weiss inequalities and the Zassenhaus Conjecture, preprint,
arxiv.org/abs/1706.02483 (2017), 21 pages.

- L. Margolis and Ã. del RÃo, Partial augmentations power property: A Zassenhaus Conjecture related problem,
preprint, arxiv.org/abs/1706.04787 (2017), 13 pages.

- L. Margolis, Ã. del RÃo, and M. Serrano, Zassenhaus conjecture on torsion units holds for
PSL(2, p) with p a Fermat or Mersenne prime, preprint, arxiv.org/abs/arXiv:1608.05797 (2016), 32 pages.

- L. Margolis and O. Sc

Final results

All work mentioned above goes beyond the state of art, mostly because a relevant conjecture that was asked formally in 1974 has been answered negatively.
The conjecture pretended to decide how are the torsion units of an integral group ring. The fact that we now know that the conjecture is false opens new possibilities for the correct answer which should be analyze in future work.
On the other hand the answer suggests ideas to to decide when the conjecture holds. This will give insize on what should be the correct answer to the main question.

By the nature of the project one cannot expect short term socio-economic and societal implications of the project but as it usually happens in mathematics the implications on long term are difficult to predict.