11.28 (1990)

Suppose the prime graph of the finite group $G$ is disconnected. (This means that the set of prime divisors of the order of $G$ is the disjoint union of non-empty subsets $\pi$ and $\pi'$ such that $G$ contains no element of order $pq$ where $p \in \pi, \ q \in \pi'$.) Then P. A. Linnell (Proc. London Math. Soc., 47, no. 1 (1983), 83–127) has proved mod CFSG that there is a decomposition of $\mathbb{Z}G$-modules $\mathbb{Z} \oplus \mathbb{Z}G = A \oplus B$ with $A$ and $B$ non-projective. Find a proof independent of CFSG.