Browsing by Subject "Fusion system"

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Embargo
On functorial equivalence classes of blocks of finite groups
(Elsevier BV * North-Holland, 2024-12) Yılmaz, Deniz
Let k be an algebraically closed field of characteristic p > 0 and let F be an algebraically closed field of characteristic 0. Recently, together with Bouc, we introduced the notion of functorial equivalences between blocks of finite groups and proved that given a p -group D , there is only a finite number of pairs ( G, b ) of a finite group G and a block b of kG with defect groups isomorphic to D , up to functorial equivalence over F. In this paper, we classify the functorial equivalence classes over F of blocks with cyclic defect groups and 2 -blocks of defects 2 and 3. In particular, we prove that for all these blocks, the functorial equivalence classes depend only on the fusion system of the block.
Open Access
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 ($A^P)^\times$.