Zassenhaus conjecture on torsion units holds for SL(2, 𝑝) and SL(2, 𝑝2)

H. J. Zassenhaus conjectured that any unit of finite order and augmentation 1 in the integral group ring {\mathbb{Z}G} of a finite group G is conjugate in the rational group algebra {\mathbb{Q}G} to an element of G. We prove the Zassenhaus conjecture for the groups {\mathrm{SL}(2,p)} and {\mathrm{SL}(2,p^{2})} with p a prime number. This is the first infinite family of non-solvable groups for which the Zassenhaus conjecture has been proved. We also prove that if {G=\mathrm{SL}(2,p^{f})} , with f arbitrary and u is a torsion unit of {\mathbb{Z}G} with augmentation 1 and order coprime with p, then u is conjugate in {\mathbb{Q}G} to an element of G. By known results, this reduces the proof of the Zassenhaus conjecture for these groups to proving that every unit of {\mathbb{Z}G} of order a multiple of p and augmentation 1 has order actually equal to p.

Microstructure and Texture Evolution of Mg–Gd–Y–Zn–Zr Alloy by Compression–Torsion Deformation

Mg–13Gd–4Y–2Zn–0.5Zr alloy was subjected to compression–torsion deformation at 450 °C with a strain rate of 0.001–0.5 s−1 using a Gleeble 3500 torsion unit. The effects of compression–torsion deformation on the microstructure and texture were studied, and the results showed that with the decrease of strain rate, the texture strength decreased, the number of dynamic precipitated particles increased, the degree of recrystallization increased, and the dynamic recrystallization mechanism changed from a continuous dynamic recrystallization mechanism to a continuous and discontinuous dynamic recrystallization mechanism. Along the direction of increasing radius, the degree of dynamic recrystallized grain (DRX) increased, the number of dynamic precipitated particles increased, and the texture strength slightly increased.

TORSION UNITS IN INTEGRAL GROUP RINGS OF CERTAIN METABELIAN GROUPS

It is shown that any torsion unit of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is rationally conjugate to an element of $\pm G$ if $G=XA$ with $A$ a cyclic normal subgroup of $G$ and $X$ an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).

On the Torsion Units of Some Integral Group Rings

It is shown that any torsion unit of the integral group ring ℤG of a finite group G is rationally conjugate to a trivial unit if G = P ⋊ A with P a normal Sylow p-subgroup of G and A an abelian p′-group (thus confirming a conjecture of Zassenhaus for this particular class of groups). The proof is an application of a fundamental result of Weiss. It is also shown that the Zassenhaus conjecture holds for PSL(2,7), the finite simple group of order 168.

Torsion Units in Integral Group Rings

Special cases of Bovdi's conjecture are proved. In particular the conjecture is proved for supersolvable and Frobenius groups. We also prove that if is finite, α ∊ VℤG a torsion unit and m the smallest positive integer such that αm ∊ G then m divides .