Splendid Morita equivalences for principal blocks with generalised quaternion defect groups

Abstract

We prove that splendid Morita equivalences between principal blocks of finite groups with dihedral Sylow 2-subgroups realised by Scott modules can be lifted to splendid Morita equivalences between principal blocks of finite groups with generalised quaternion Sylow 2-subgroups realised by Scott modules.

The Brauer indecomposability of Scott modules with vertex $Q_{2^n}\times C_{2^m}$

“…Then the Scott module Sc(G × G ′ , ∆P ) is Brauer indecomposable. This result generalizes Lemma 2.2 of [14]. The paper is divided into four sections.…”

“…The goal of this paper is to prove that the Scott module whose vertex is isomorphic to a direct product of a generalized quaternion 2-group and a cyclic 2-group is Brauer indecomposable. The Brauer indecomposability of Scott modules is an important notion because it serves a key ingredient for the Scott module to realize a splendid Morita equivalence between certain principal blocks with isomorphic defect groups (see [11][12][13][14][15][16][17]22]).…”

Character triples and equivalences over a group graded G-algebra

“…We also prove in Theorem 3.9 an analogue of the Butterfly theorem [15,Theorem 2.16], generalizing the main result of [12]. In Section 4 we show how to obtain Ḡ-graded Morita equivalences over C from the Morita equivalences induced by the Scott module Sc(N × N ′ , ∆Q) of Koshitani and Lassueur [7], [8].…”

“…Proof. Consider the following diagram: 4 Scott modules Koshitani and Lassueur constructed in [7] and [8] Morita equivalences induced by certain Scott modules. We show here that their constructions can be extended to obtain group graded Morita equivalences over C = OC G (N ).…”

Brauer indecomposability of Scott modules with semidihedral vertex

“…These theorems in a sense generalize [11][12][13][14], and there are results on Brauer indecomposability of Scott modules also in [15,21]. Notation 1.3.…”