Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 2, pp. 513-520 (2002)

Another Counterexample to a Conjecture of Zassenhaus

M. Hertweck

Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany, e-mail:

Abstract: A metabelian group $G$ of order 1440 is constructed which provides a counterexample to a conjecture of Zassenhaus on automorphisms of integral group rings. The group is constructed in the spirit of [8]. An augmented automorphism of ${\hbox{$\Bbb Z$}G$ which has no Zassenhaus factorization is given explicitly (this was already done in [7] for a group of order $6720$), but this time only a few distinguished group ring elements are used for its construction, carefully exploiting certain congruence relations satisfied by powers of these elements.

[7] Klingler, L.: Construction of a counterexample to a conjecture of Zassenhaus. Comm. Algebra 19 (1991), 2303-2330.

[8] Roggenkamp, K. W.; Scott, L. L.: On a conjecture of Zassenhaus for finite group rings. Manuscript, November 1988, 1-60.

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