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Kimmerle conjecture for the Held and O’Nan sporadic simple groups. (English) Zbl 1182.16030

A long standing Zassenhaus conjecture says that every normalized torsion unit of the integral group ring \(\mathbb{Z} G\) of a finite group \(G\) is conjugated, within the rational group algebra \(\mathbb{Q} G\), to an element in \(G\).
A weakened version of this conjecture concerns the Gruenberg-Kegel graph (also called the prime graph) \(\pi(X)\) of an arbitrary group \(X\). Recall that the vertices of this graph are labeled by primes \(p\) for which there exists an element of order \(p\) in \(X\) and with an edge from \(p\) to a distinct \(q\) if \(X\) has an element of order \(pq\). The following question was posed by W. Kimmerle: is \(\pi(V(\mathbb{Z} G))=\pi(G)\), for a finite group \(G\)? Of course, a positive answer to the Zassenhaus conjecture implies a positive answer to Kimmerle’s question. Positive answers to the latter have been given by Kimmerle for finite Frobenius and solvable groups, and in a series of papers, by Bovdi and Konovalov and also Hertweck, Jespers, Linton, Marcos, Siciliano, for some simple groups, including 12 of the 26 sporadic simple groups.
In this paper a positive answer to Kimmerle’s question is proved for the Held sporadic group. For the O’Nan sporadic simple group the non-existence of torsion units of all relevant orders, except orders 33 and 57, is given. For both groups, some extra information is obtained that is relevant for the Zassenhaus conjecture.

MSC:

16U60 Units, groups of units (associative rings and algebras)
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
20D08 Simple groups: sporadic groups

Software:

LAGUNA; GAP
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