The group of splendid Morita equivalences of principal 2-blocks with dihedral and generalised quaternion defect groups

Authors : Çisil Karagüzel, Deniz Yilmaz

Pages : 160-167

Doi:10.24330/ieja.1402947

View : 102 | Download : 58

Publication Date : 2024-01-09

Article Type : Research Paper

Abstract :Let $k$ be an algebraically closed field of characteristic $2$, let $G$ be a finite group and let $B$ be the principal $2$-block of $kG$ with a dihedral or a generalised quaternion defect group $P$. Let also $\\calT(B)$ denote the group of splendid Morita auto-equivalences of $B$. We show that \\begin{align*} \\calT(B)\\cong \\Out_P(A)\\rtimes \\Out(P,\\calF), \\end{align*} where $\\Out(P,\\calF)$ is the group of outer automorphisms of $P$ which stabilize the fusion system $\\calF$ of $G$ on $P$ and $\\Out_P(A)$ is the group of algebra automorphisms of a source algebra $A$ of $B$ fixing $P$ modulo inner automorphisms induced by $(A^P)^\\times$.
Keywords : Block, fusion system, Picard group, dihedral defect group, generalised quaternion defect group

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