Clifford theory of characters in induced blocks

Abstract

We present a new criterion to predict if a character of a finite group extends. Let $G$ be a finite group and $p$ a prime. For $N⊲G$, we consider $p$-blocks $b$ and $b^{\prime }$ of $N$ and ${\mathrm{N}}_N(D)$, respectively, with $(b^{\prime }{)}^N=b$, where $D$ is a defect group of $b^{\prime }$. Under the assumption that $G$ coincides with a normal subgroup $G[b]$ of $G$, which was introduced by Dade early 1970's, we give a character correspondence between the sets of all irreducible constituents of ${\varphi }^G$ and those of $({\varphi }^{\prime }{)}^{{\mathrm{N}}_G(D)}$ where $\varphi$ and ${\varphi }^{\prime }$ are irreducible Brauer characters in $b$ and $b^{\prime }$, respectively. This implies a sort of generalization of the theorem of Harris-Knörr. An important tool is the existence of certain extensions that also helps in checking the inductive Alperin-McKay and inductive Blockwise Alperin Weight conditions, due to the second author.

References in corpus (1)

• The inductive Alperin-McKay and blockwise Alperin weight conditions for blocks with cyclic defect groups