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 :
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 . 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, ). One of the most interesting modifications was suggested by W. Kimmerle . Let us briefly introduce it now.
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