A generalization of Alperin fusion theorem and its applications

Abstract

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.