We give a sufficient condition for the kG-Scott module with vertex P to remain indecomposable under taking 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 wreathed 2subgroup of a finite group G. This generalizes results for the cases where P is abelian and some others. The motivation of this paper is that the Brauer indecomposability of a p-permutation bimodule (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 that then can possibly lift to a splendid derived equivalence.
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.
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