Normality in group rings

Let $KG$ be the group ring of a group $G$ over a commutative ring $K$ with unity. The rings $KG$ are described for which $xx^\sigma=x^\sigma x$ for all $x=\sum_{g\in G}\alpha_gg\in KG$, where \quad $x\mapsto x^\sigma=~\sum_{g\in G}\alpha_gf(g)\sigma(g)$\quad is an involution of $KG$; here $f: G\to U(K)$ is a homomorphism and $\sigma$ is an anti-automorphism of order two of $G$.Comment: 8 page

Kimmerle conjecture for the Held and O'Nan sporadic simple groups

Using the Luthar--Passi method, we investigate the Zassenhaus and Kimmerle conjectures for normalized unit groups of integral group rings of the Held and O'Nan sporadic simple groups. We confirm the Kimmerle conjecture for the Held simple group and also derive for both groups some extra information relevant to the classical Zassenhaus conjecture.Comment: 9 page

Torsion Units for a Ree group, Tits group and a Steinberg triality group

We investigate the Zassenhaus conjecture for the Steinberg triality group ${}^3D_4(2^3)$, Tits group ${}^2F_4(2)'$ and the Ree group ${}^2F_4(2)$. Consequently, we prove that the Prime Graph question is true for all three groups