Phantom maps and purity in modular representation theory, III D. J. Benson Department of Mathematics, University of Georgia, Athens GA 30602, USA Abstract: In the third installment of this paper, we exhibit two phantom maps between $kG$-modules, whose composite is not projective, whenever $k$ is an uncountable field of characteristic $p$ and $G$ is a finite group of $p$-rank at least two.