, Volume 16, Issue 6, pp 1717-1732
Date: 25 Sep 2012

Blocks with Defect Group \(\boldsymbol{Q_{2^n}\times C_{2^m}}\) and \(\boldsymbol{SD_{2^n}\times C_{2^m}}\)

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Abstract

We determine the numerical invariants of blocks with defect group \(Q_{2^n}\times C_{2^m}\) and \(SD_{2^n}\times C_{2^m}\) , where \(Q_{2^n}\) denotes a quaternion group of order 2 n , \(C_{2^m}\) denotes a cyclic group of order 2 m , and \(SD_{2^n}\) denotes a semidihedral group of order 2 n . This generalizes Olsson’s results for m = 0. As a consequence, we prove Brauer’s k(B)-Conjecture, Olsson’s Conjecture, Brauer’s Height-Zero Conjecture, the Alperin–McKay Conjecture, Alperin’s Weight Conjecture and Robinson’s Ordinary Weight Conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper follows (and uses) (Sambale, J Pure Appl Algebra 216:119–125, 2012; Proc Amer Math Soc, 2012).

Presented by Alain Verschoren.