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Published online by Cambridge University Press: 07 May 2010
Abstract
We investigate the classical Zassenhaus conjecture for the normalized unit group of the integral group ring of the simple Mathieu group M11. As a consequence, for this group we confirm the conjecture by Kimmerle about prime graphs.
Introduction and main results
Let V (ℤG) be the normalized unit group of the integral group ring ℤG of a finite group G. The following famous conjecture was formulated by H. Zassenhaus in [15]:
Conjecture 1 (ZC) Every torsion unit u ∈ V(ℤG) is conjugate within the rational group algebra ℚG to an element of G.
This conjecture is already confirmed for several classes of groups but, in general, the problem remains open, and a counterexample is not known.
Various methods have been developed to deal with this conjecture. One of the original ones was suggested by I. S. Luthar and I. B. S. Passi [12, 13], and it was improved further by M. Hertweck [9]. Using this method, the conjecture was proved for several new classes of groups, in particular for S5 and for some finite simple groups (see [4, 9, 10, 12, 13]).
The Zassenhaus conjecture appeared to be very hard, and several weakened variations of it were formulated (see, for example, [3]). One of the most interesting modifications was suggested by W. Kimmerle [11]. Let us briefly introduce it now.
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