论文标题
$ p $ - 最大级别的饱和融合系统
Saturated fusion systems on $p$-groups of maximal class
论文作者
论文摘要
对于Prime Number $ P $，有限的$ P $ - 订单$ P^n $具有最大类，如果它具有Nilpotency类$ N-1 $。在这里，我们检查了Maximal类$ P $ - 组上的饱和融合系统，尤其是我们描述了所有重新数字$ P $，有限的$ p $ - 订单$ p^n $具有最大级别，并且仅当它具有nilpotency class $ n-1 $。在这里，我们检查了最大级别$ p $ -groups $ s $ s $至少$ p^4 $的饱和融合系统$ \ MATHCAL F $。饱和融合系统的Alperin-Goldschmidt定理产生的是，$ \ Mathcal F $完全取决于其$ \ MATHCAL F $ - ESSEALTIAN SUBGROUPS和$ S $本身的$ \ Mathcal F $ -Automorphisms。如果$ \ MATHCAL F $ - ESTENTAILAL subgroup具有订单$ p^2 $，或者是订单$ p^3 $的非亚洲，则称为$ \ MATHCAL F $ -PEARL。这项工作中的促进和技术定理表明，$ \ Mathcal F $ - ESTENTAILAL子组是$ \ Mathcal F $ -PEARL，或两个明确确定的最大亚组之一的$ S $。如果$ s $是$ 2 $ group，则可以轻松证明此结果，并且可以从D'\ iaz，Ruiz和Viruel的工作中阅读，而当$ p = 3 $时，可以与Parker和Semeraro一起阅读。主要贡献是$ p \ ge 5 $，因为在这种情况下，最大级别$ p $ groups没有分类。 主要定理描述了最大级别$ p $ - 至少$ p^4 $的所有减少饱和融合系统，并从另外两个广泛的定理中进行了。这两个定理描述了所有饱和融合系统，而不是分别限制在降低的融合系统上，例如分别在特殊和非远程最大类$ p $ group上。作为推论，我们很容易记住的结果，表明，如果$ o_p（\ mathcal f）= 1 $，那么$ \ mathcal f $具有$ \ MATHCAL F $ -PEARLS或$ S $对SYLOW $ P $ -P $ -SUBGROUP的$ \ \ Mathrm G_2（p）$ $ $ $ pe \ ge 5 \ ge 5 \ 5 \ ge 5 \ ge 5.sylow $ p $ p $ p $＆p $。
For a prime number $p$, a finite $p$-group of order $p^n$ has maximal class if it has nilpotency class $n-1$. Here we examine saturated fusion systems on maximal class $p$-groups and, in particular, we describe all the reduFor a prime number $p$, a finite $p$-group of order $p^n$ has maximal class if and only if it has nilpotency class $n-1$. Here we examine saturated fusion systems $\mathcal F$ on maximal class $p$-groups $S$ of order at least $p^4$. The Alperin-Goldschmidt Theorem for saturated fusion systems yields that $\mathcal F$ is entirely determined by the $\mathcal F$-automorphisms of its $\mathcal F$-essential subgroups and of $S$ itself. If an $\mathcal F$-essential subgroup either has order $p^2$ or is non-abelian of order $p^3$, then it is called an $\mathcal F$-pearl. The facilitating and technical theorem in this work shows that an $\mathcal F$-essential subgroup is either an $\mathcal F$-pearl, or one of two explicitly determined maximal subgroups of $S$. This result is easy to prove if $S$ is a $2$-group and can be read from the work of D'\iaz, Ruiz, and Viruel together with that of Parker and Semeraro when $p=3$. The main contribution is for $p \ge 5$ as in this case there is no classification of the maximal class $p$-groups. The main Theorem describes all the reduced saturated fusion systems on a maximal class $p$-group of order at least $p^4$ and follows from two more extensive theorems. These two theorems describe all saturated fusion systems, not restricting to the reduced ones for example, on exceptional and non-exceptional maximal class $p$-groups respectively. As a corollary, we have the easy to remember result that states that, if $O_p(\mathcal F)=1$, then either $\mathcal F$ has $\mathcal F$-pearls or $S$ is isomorphic to a Sylow $p$-subgroup of $\mathrm G_2(p)$ with $p\ge 5$ and the fusion systems are explicitly described.
