Decomposition numbers of $p$-solvable groups

In the character theory of finite groups one decomposes each ordinary irreducible character ${\chi _i}$ of a group into an integral linear combination of $p$-modular irreducible characters ${\phi _j},{\chi _i} = \sum {{d_{ij}}{\phi _j}}$. The nonnegative integers ${d_{ij}}$ are called the $p$-decomposition numbers. Let $G$ be a $p$-solvable group whose $p$-Sylow subgroups are abelian. If $G/{O_{p’p}}(G)$ is cyclic the $p$-decomposition numbers are $\leqq 1$. This condition is far from necessary as any group $G$ with abelian, normal $p$-Sylow subgroup $P$ with $G/P$ abelian has $p$-decomposition numbers $\leqq 1$. A result of Brauer and Nesbitt together with the first result yields the following. A group $G$ has a normal $p$-complement and abelian $p$-Sylow subgroups if and only if each irreducible character of $G$ is irreducible as a $p$-modular character.