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- W2098046975 abstract "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 $G/{O_{pâp}}(G)$ 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." @default.
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- W2098046975 date "1970-01-01" @default.
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- W2098046975 title "Decomposition numbers of $p$-solvable groups" @default.
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- W2098046975 doi "https://doi.org/10.1090/s0002-9939-1970-0254146-2" @default.
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