Torsion units in integral group rings of some metabelian groups. II
This continuation of the work on the Zassenhaus conjecture [see Part I by the two last named authors, J. Algebra 103, 490-499 (1986; Zbl 0603.16010)] uses an entirely new approach which seems to be very interesting and, in the opinion of the authors, might lead to a proof of the Zassenhaus conjecture for groups G of the form \(A\rtimes X\) with A and X abelian and of relatively prime order. Instead of using explicit computations of the representations of G, the well known embedding \(ZG\to (ZA)_ m\) (m\(\times m\) matrices over ZA) is studied. For better application of this context, it is shown that the Zassenhaus conjecture is equivalent to the following statement: ''For every \(u=\sum u_ gg\), a torsion, normalized unit of ZG, there exists an element \(g_ 0\) of G, unique up to conjugation, such that the sum of the coefficients \(u_ g\) of the element g conjugate to \(g_ 0\) is different from zero''. With this new technique, the Zassenhaus conjecture is proved when the order of X, m, is less than the primes dividing the order of A in the following special cases: 1) m is prime; 2) A is cyclic.
scientific article; zbMATH DE number 3987492
scientific article