Speaker: Matthew Gelvin (Bilkent University, Ankara) Title:Fusionminimal groups Place: Room Seminar C3b Date: Friday April 27, 12h13h
Abstract: Every saturated fusion system $\mathcal{F}$ on the $p$group $S$ has an associated collection of characteristic bisets. These are $(S,S)$bisets that determine $\mathcal{F}$, and are in turn determined by $\mathcal{F}$ up to a moreorless explicit parameterization. In particular, there is always a unique minimal $\mathcal{F}$characteristic biset, $\Omega_\mathcal{F}$. If $G$ is a finite group containing $S$ as a Sylow $p$subgroup and realizing $\mathcal{F}$, then $G$ is itself, when viewed as an $(S,S)$biset, $\mathcal{F}$characteristic. If it happens that $_SG_S=\Omega_\mathcal{F}$ is the minimal biset for its fusion system, we say that $G$ is \emph{fusionminimal}.
In joint work with Sune Reeh, it was shown that any strictly $p$constrained group (i.e., one that satisfies $C_G(O_p(G))\leq O_p(G)$) is fusion minimal. We conjecture that converse implication holds. In this talk, based on joint work with Justin Lynd, we prove this to be the case when $p$ is odd and describe the obstruction to a complete proof. Along the way, we will draw a connection with the module structure of block algebras and how this relates to the question at hand.
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