Torsion units in integral group rings

Several special cases of the conjectures of Bovdi and Zassenhaus are proved. We also deal with special cases of the following conjecture: let α be a torsion unit of the integral group ring ZZG and m the smallest positive integer such that α^m ∈G then, m is a divisor of the exponent of the quotient group G/Z(G) provided this exponent is finite