Saturated fusion systems on p-groups of maximal class

For a prime number p, a finite p-group of order pn 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 pn has maximal class if and only if it has nilpotency class n−1. Here we examine saturated fusion systems F on maximal class p-groups S of order at least p4. The Alperin-Goldschmidt Theorem for saturated fusion systems yields that F is entirely determined by the F-automorphisms of its F-essential subgroups and of S itself. If an F-essential subgroup either has order p2 or is non-abelian of order p3, then it is called an F-pearl. The facilitating and technical theorem in this work shows that an F-essential subgroup is either an 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≥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 p4 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 Op(F)=1, then either F has F-pearls or S is isomorphic to a Sylow p-subgroup of G2(p) with p≥5 and the fusion systems are explicitly described.