论文标题
Alperin Fusion定理及其应用的概括
A generalization of Alperin fusion theorem and its applications
论文作者
论文摘要
令$ \ Mathcal F $为有限$ P $ -Group $ S $上的饱和融合系统，让$ p $成为强烈的$ \ \ \ natercal f $ f $ clupsed $ s $的子组。我们定义了``$ \ natercal f $ f $ - 基本的子组相对于$ p $''，这是满足某些技术条件的$ p $的某些适当亚组，并表明$ p $ $ p $之间的$ p $ $ p $之间的$ p $ $ p $ p $ p $ p $ p $ $ p $ $ $ $ $ $ \ $ $ $ $ p $之间的$ \ \ \ \ \ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\不然$ p $是相等的$ s $，Alperin-goldschmidt融合定理可以作为一种特殊情况，我们还可以证明$ p \ limhd \ mathcal f $，并且只有在没有$ p $ p $的情况下，只有$ \ mathcal f $ exteriality of $ p $。 \ Mathcal f $每当$ p $ -group $ s $和饱和的融合系统上的$ \ Mathcal f $上的$ s $，因此$ p $强烈\ \ mathcal f $ f $ clused $ \ p $ cluct $ \ p $ cluct $ p $ clups of for $ p $。在$ \ Mathcal F $中，子组是正常的。
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.
