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 $(A^P)^\times$.
📌 3 Key Takeaways
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Objective & Hypothesis: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 $P$.
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Key Findings & Data:Let also $\calT(B)$ denote the group of splendid Morita auto-equivalences of $B$.
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Conclusion & Impact: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 $(A^P)^\times$.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Block · fusion system · Picard group
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 $(A^P)^\times$.
Karagüzel, Ç., Yılmaz, D. (2024). The group of splendid Morita equivalences of principal 2-blocks with dihedral and generalised quaternion defect groups. International Electronic Journal of Algebra, Volume: 35 Issue: 35. https://doi.org/10.24330/ieja.1402947
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