Browsing by Subject "Dihedral defect group"

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    The group of splendid morita equivalences of principal 2-blocks with dihedral and generalised quaternion defect groups
    (International Electronic Journal of Algebra, 2024-01-09) Karagüzel, Çisil; Yılmaz, Deniz
    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 ($A^P)^\times$.