Saturated Fusion Systems on $p$-Groups of Maximal Class

Valentina Grazian, Universita degli Studi di Padova, Italy. Chris Parker, University of Birmingham, United Kingdom.

Chapters 1. Introduction and main results 2. General group theoretical results 3. Maximal class $p$-groups 4. Representations of groups with a cyclic Sylow $p$-subgroup 5. A primer on fusion systems 6. Fusion systems on $p$-groups of maximal class: Generalities 7. Essential subgroups in exceptional maximal class $p$-groups 8. The structure of $\gamma _1(S)$ when $\gamma _1(S)$ is $\mathcal {F}$-essential and $S$ is not exceptional 9. The structure of $\gamma _1(S)$ when $\gamma _1(S)$ is $\mathcal {F}$-essential and $S$ is exceptional 10. Locating $\mathcal {F}$-essential subgroups in groups of maximal class I 11. Locating $\mathcal {F}$-essential subgroups in groups of maximal class II 12. The saturated fusion systems on exceptional maximal class groups: The proof of Theorem B 13. The saturated fusion systems on non-exceptional maximal class groups 14. The proofs of Theorems A and C 15. A series of examples with non-abelian $2$-step centralizer A. Saturated fusion systems on maximal class $p$-groups with $\gamma _1(S)$ abelian B. The saturated fusion systems on maximal class $3$-groups C. Computer code