Overgroups of Root Groups in Classical Groups

Michael Aschbacher, Caltech, Pasadena, CA, USA.

• Introduction 3-transpositionsThe $(V,f)$-setupDirect sum decompositionsSubfield structuresModules for alternating groupsModules with $p=2$The orthogonal space $\mathbf{F}_2^n$Overgroups of long root subgroupsMaximal overgroups of long root subgroupsSubgroups containing long root elementsOvergroups of short root subgroupsShort root subgroups in symplectic groups of characteristic 2Overgroups of subgroups in $\mathbf{R}_c$ in IIIOvergroups of subgroups in $\mathbf{R}_c$ in III when $q>3$A special case for $q=3$ in IIIOvergroups of subgroups in $\mathbf{R}_c$ in III when $q=3$A result of Stellmacher More case III with $q=3$The proof of Theorem 1A characterization of alternating groupsOrthogonal groups with $q=2$The proof of Theorem 2Symplectic and unitary groupsSymplectic and unitary groups with $q$ oddThe proof of Theorem 3Unitary groups with $q$ evenThe proofs of Theorems A and BReferences