2022/04/24 by Kızmaz, M. Yasir
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2204.11303
Let \mathcal F be a saturated fusion system on a finite p-group S, and let P be a strongly \mathcal F-closed subgroup of S. We define the concept ``\mathcal F-essential subgroups with respect to P" which are some proper subgroups of P satisfying some technical conditions, and show that an \mathcal F-isomorphism between subgroups of P can be factorised by some automorphisms of P and \mathcal F-essential subgroups with respect to P. When P is taken to be equal S, Alperin-Goldschmidt fusion theorem can be obtained as a special case. We also show that P\unlhd \mathcal F if and only if there is no \mathcal F-essential subgroup with respect to P. The following definition is made: a p-group P is strongly resistant in saturated fusion systems if P\unlhd \mathcal F whenever there is an over p-group S and a saturated fusion system \mathcal F on S such that P is strongly \mathcal F-closed. It is shown that several classes of p-groups are strongly resistant, which appears as our third main theorem. We also give a new necessary and sufficient criteria for a strongly \mathcal F-closed subgroup to be normal in \mathcal F. These results are obtained as a consequences of developing a theory of quasi and semi-saturated fusion systems, which seems to be interesting for its own right.