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 Z[Q 8 × 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 ≡ 3 (mod 4). We also prove that Z[Q 8 × C 5 ] has the multiplicative Jordan decomposition property in a new way.