The Multiplicative Jordan Decomposition in the Integral Group Ring $\mathbb{Z}[Q_8\times C_p]$

Abstract

Let $p$ be a prime such that the multiplicative order $m$ of $2$ modulo $p$ is even. We prove that the integral group ring $\mathbb{Z}[Q_8\times C_p]$ has the multiplicative Jordan decomposition property when $m$ is congruent to $2$ modulo $4$. There are infinitely many such primes and these primes include the case $p\equiv 3(\mathrm{mod}4)$. We also prove that $\mathbb{Z}[Q_8\times C_5]$ has the multiplicative Jordan decomposition property in a new way.