We present a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a semidihedral $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian or dihedral. The Brauer indecomposability is defined by R. Kessar, N. Kunugi and N. Mitsuhashi. The motivation of this paper is the fact that the Brauer indecomposability of a $p$-permutation bimodule (where $p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method due to Broué, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.

中文翻译：

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具有半二面体顶点的 Scott 模的 Brauer 不可分解性

我们提出了一个充分条件$kG$- 带顶点的斯科特模块$P$在任何子群的布劳尔构造下保持不可分解$Q$的$P$作为$k[Q\,C_G(Q)]$-模块，其中$k$是一个特征领域$2$， 和$P$是一个半二面体$2$-有限群的子群$G$. 这概括了以下情况的结果$P$是阿贝尔或二面角。Brauer 不可分解性由 R. Kessar、N. Kunugi 和 N. Mitsuhashi 定义。本文的动机是一个事实，即 Brauer 不可分解$p$-排列双模（其中$p$是一个素数）是通过使用 Broué、Rickard、Linckelmann 和 Rouquier 的粘合方法获得 Morita 类型的出色稳定等效项的关键步骤之一，然后可以提升到出色的派生（出色森田）等价。