The conjecture of Zassenhaus about torsion units of group rings claims that, given the finite group G and a torsion unit u of ZG, there exists a unit v of QG such that \(vuv^{-1}\) is an element of G. So far, the conjecture was known to be true for groups which are nilpotent of class two, for certain metacyclic groups and for dihedral groups [see the survey of C. Polcino Milies in ”Group and semigroup rings”, North- Holland Math. Stud. 126, 179-192 (1986; Zbl 0596.16006) and T. Mitsuda in Commun. Algebra 14, 1707-1728 (1986; Zbl 0602.16008)].
The present paper extends the validity of the conjecture to extensions \(0\to A\to G\to X\to 1\) where A is an elementary abelian p-group and X is an abelian group with faithful, irreducible action on A. The techniques which are needed for the proof follow the ones that were introduced in previous works on the subject by C. Polcino Milies, J. Ritter and S. K. Sehgal, but now some sophisticated computations are used, based on the special structure of G. In particular, two families of operators are introduced: \(\bar S{}_{\theta}(\delta)\) and \(Y_{\theta}(\delta)\), where the indices \(\delta\) run over \(\Delta (G,A)/\Delta (G,A)^ 2\) and where \(\bar S{}_{\theta}(\delta)\), \(Y_{\theta}(\delta)\) are K- endomorphisms of the group ring KX (K being the finite field of \(| A|\) elements). The authors show some interesting relations between these operators which allow to prove that, given the unit \(u=1+\delta\), there is an \(a\in A\) such that \(Y_{\theta}(\delta)\) and \(Y_{\theta}(a-1)\) have the same eigenvalues. From this, it is not hard to complete the desired proof.