Saturated fusion systems on $p$-groups of maximal class

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.