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

• Chapters1. Introduction and main results2. General group theoretical results3. Maximal class $p$-groups4. Representations of groups with a cyclic Sylow $p$-subgroup5. A primer on fusion systems6. Fusion systems on $p$-groups of maximal class: Generalities7. Essential subgroups in exceptional maximal class $p$-groups8. The structure of $\gamma _1(S)$ when $\gamma _1(S)$ is $\mathcal {F}$-essential and $S$ is not exceptional9. The structure of $\gamma _1(S)$ when $\gamma _1(S)$ is $\mathcal {F}$-essential and $S$ is exceptional10. Locating $\mathcal {F}$-essential subgroups in groups of maximal class I11. Locating $\mathcal {F}$-essential subgroups in groups of maximal class II12. The saturated fusion systems on exceptional maximal class groups: The proof of Theorem B13. The saturated fusion systems on non-exceptional maximal class groups14. The proofs of Theorems A and C15. A series of examples with non-abelian $2$-step centralizerA. Saturated fusion systems on maximal class $p$-groups with $\gamma _1(S)$ abelianB. The saturated fusion systems on maximal class $3$-groupsC. Computer code

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

• Chapters1. Introduction and main results2. General group theoretical results3. Maximal class $p$-groups4. Representations of groups with a cyclic Sylow $p$-subgroup5. A primer on fusion systems6. Fusion systems on $p$-groups of maximal class: Generalities7. Essential subgroups in exceptional maximal class $p$-groups8. The structure of $\gamma _1(S)$ when $\gamma _1(S)$ is $\mathcal {F}$-essential and $S$ is not exceptional9. The structure of $\gamma _1(S)$ when $\gamma _1(S)$ is $\mathcal {F}$-essential and $S$ is exceptional10. Locating $\mathcal {F}$-essential subgroups in groups of maximal class I11. Locating $\mathcal {F}$-essential subgroups in groups of maximal class II12. The saturated fusion systems on exceptional maximal class groups: The proof of Theorem B13. The saturated fusion systems on non-exceptional maximal class groups14. The proofs of Theorems A and C15. A series of examples with non-abelian $2$-step centralizerA. Saturated fusion systems on maximal class $p$-groups with $\gamma _1(S)$ abelianB. The saturated fusion systems on maximal class $3$-groupsC. Computer code

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

• Chapters1. Introduction and main results2. General group theoretical results3. Maximal class $p$-groups4. Representations of groups with a cyclic Sylow $p$-subgroup5. A primer on fusion systems6. Fusion systems on $p$-groups of maximal class: Generalities7. Essential subgroups in exceptional maximal class $p$-groups8. The structure of $\gamma _1(S)$ when $\gamma _1(S)$ is $\mathcal {F}$-essential and $S$ is not exceptional9. The structure of $\gamma _1(S)$ when $\gamma _1(S)$ is $\mathcal {F}$-essential and $S$ is exceptional10. Locating $\mathcal {F}$-essential subgroups in groups of maximal class I11. Locating $\mathcal {F}$-essential subgroups in groups of maximal class II12. The saturated fusion systems on exceptional maximal class groups: The proof of Theorem B13. The saturated fusion systems on non-exceptional maximal class groups14. The proofs of Theorems A and C15. A series of examples with non-abelian $2$-step centralizerA. Saturated fusion systems on maximal class $p$-groups with $\gamma _1(S)$ abelianB. The saturated fusion systems on maximal class $3$-groupsC. Computer code