Serwene's realizability conjecture for saturated fusion systems

A saturated fusion system F\mathcal{F} on a finite pp-group SS is realizable if there is a finite group G1G_1 such that F=FS(G1)\mathcal{F}=\mathcal{F}_{S}(G_1), where SS is a Sylow pp-subgroup of G1G_1; otherwise, F\mathcal{F} is exotic. An F\mathcal{F}-block of a finite group GG is a block bb associated with the saturated fusion system F\mathcal{F}.

Serwene's conjecture. Let F\mathcal{F} be a saturated fusion system. For any finite group GG having an F\mathcal{F}-block bb, it follows that F\mathcal{F} is realizable.

This conjecture concerns whether saturated fusion systems arising from blocks of finite groups must be realizable rather than exotic. The supplied text gives no resolution status beyond stating it as a conjecture.

Sources & referencesView supporting material

Primary source

Patrick Serwene and Constantin-Cosmin Todea, “A reduction theorem for non-vanishing of Hochschild cohomology of block algebras and Happel's property”, arXiv:2503.05432 (2025).

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