The influence of minimal subgroups on saturated fusion systems


Let \cal F be a saturated fusion system over a \py group S. In this paper, we investigate the influence of the minimal subgroups in \mathfrak{foc}(\mathcal{F}), the focal subgroup of \cal F. Our main result is that if for each cyclic subgroup P\leq \mathfrak{foc}(\mathcal{F}) of order p (of order 2 and 4 if p=2) and each \varphi\in \sHom_{\cal F}(P,\mathfrak{foc}(\mathcal{F})), \varphi extends to an automorphism of S, then S is normal in \cal F. We also give several applications of this result

DOI Code: 10.1285/i15900932v36suppl1p91

Keywords: saturated fusion system; focal subgroup; resistant group

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