University of Birmingham > Talks@bham > Algebra Seminar > Torsion Units in Integral Group Rings of Sporadic Simple Groups

Torsion Units in Integral Group Rings of Sporadic Simple Groups

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If you have a question about this talk, please contact Simon Goodwin.

(joint work with Victor Bovdi, Eric Jespers, Steve Linton, Salvatore Siciliano et al.)

Let U(ZG) be the unit group of the integral group ring ZG of a finite group G. A unit of ZG of the form a1g1+a2g2+···+angn with ai in Z and gi in G is called normalised, if a1+a2+···+an = 1. Normalised units form a subgroup of U(ZG) called the normalized unit group of ZG and denoted by V(ZG).

The long-standing conjecture of H.Zassenhaus (ZC) says that every torsion unit from V(ZG) is conjugate within the rational group algebra QG to an element of G. One of its weakened variations can be formulated in terms of the Gruenberg-Kegel graph (also called the prime graph) of an arbitrary group X, which has vertices labeled by primes p for which there exists an element of order p in X and edges between distinct primes p and q if and only if X has an element of order pq. Clearly, if (ZC) holds for a finite group G, then G and V(ZG) have the same prime graph.

The criterion for ZC can be formulated in terms of vanishing of partial augmentations of torsion units (for an element of a group ring of the form a1g1+a2g2+···+angn with ai in Z and gi in G, its partial augmentation with respect to the conjugacy class C of elements of the group G is the sum of coefficients ai over those gi which belong to the class C). Therefore, it is useful to know for each possible order of a torsion unit in V(ZG), which combinations of partial augmentations may arise.

This motivated us to start the project to collect information about possible partial augmentations of torsion units of integral group rings of sporadic simple groups. As a consequence, at the time of this talk we proved that G and V(ZG) have the same prime graph for the following thirteen sporadic simple groups:

  • Mathieu groups M11, M12, M22, M23, M24;
  • Janko groups J1, J2, J3;
  • Held, Higman-Sims, McLaughlin, Rudvalis and Suzuki groups.

In my talk I will summarise known information about orders and partial augmentations for these groups, explain enhancements of the Luthar-Passi method that were developed during the project, and highlight some challenges arising from the remaining sporadic simple groups.

This talk is part of the Algebra Seminar series.

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