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# Barcelona Algebraic Topology Group

Barcelona Algebraic Topology Group
 Friday's Topology Seminar 2018-19
 Written by Natàlia Castellana Vila Friday, 30 November 2018 14:12 Speaker: Matthew Gelvin (Bilkent University, Ankara)Title:Fusion-minimal groups Place: Room Seminar C3bDate: Friday April 27, 12h-13h 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 more-or-less 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{fusion-minimal}.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. See the calendar for upcoming events. Last Updated on Tuesday, 23 April 2019 13:11 Read more...

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