越谷 重夫

<title>Abstract</title> We present a sufficient condition for the <inline-formula> <alternatives> <tex-math>$kG$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline1.png" /> </alternatives> </inline-formula>-Scott module with vertex <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline2.png" /> </alternatives> </inline-formula> to remain indecomposable under the Brauer construction for any subgroup <inline-formula> <alternatives> <tex-math>$Q$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline3.png" /> </alternatives> </inline-formula> of <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline4.png" /> </alternatives> </inline-formula> as <inline-formula> <alternatives> <tex-math>$k[Q\,C_G(Q)]$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline5.png" /> </alternatives> </inline-formula>-module, where <inline-formula> <alternatives> <tex-math>$k$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline6.png" /> </alternatives> </inline-formula> is a field of characteristic <inline-formula> <alternatives> <tex-math>$2$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline7.png" /> </alternatives> </inline-formula>, and <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline8.png" /> </alternatives> </inline-formula> is a semidihedral <inline-formula> <alternatives> <tex-math>$2$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline9.png" /> </alternatives> </inline-formula>-subgroup of a finite group <inline-formula> <alternatives> <tex-math>$G$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline10.png" /> </alternatives> </inline-formula>. This generalizes results for the cases where <inline-formula> <alternatives> <tex-math>$P$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline11.png" /> </alternatives> </inline-formula> 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 <inline-formula> <alternatives> <tex-math>$p$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline12.png" /> </alternatives> </inline-formula>-permutation bimodule (where <inline-formula> <alternatives> <tex-math>$p$</tex-math> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0013091521000067_inline13.png" /> </alternatives> </inline-formula> 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.