Overgroups of root groups in classical groups / Michael Aschbacher.

Includes bibliographical references.

* Introduction 3-transpositions* The $(V,f)$-setup* Direct sum decompositions* Subfield structures* Modules for alternating groups* Modules with $p=2$* The orthogonal space $\mathbf{F}_2^n$* Overgroups of long root subgroups* Maximal overgroups of long root subgroups* Subgroups containing long root elements* Overgroups of short root subgroups* Short root subgroups in symplectic groups of characteristic 2* Overgroups of subgroups in $\mathbf{R}_c$ in III* Overgroups of subgroups in $\mathbf{R}_c$ in III when $q>3$* A special case for $q=3$ in III* Overgroups 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 1* A characterization of alternating groups* Orthogonal groups with $q=2$* The proof of Theorem 2* Symplectic and unitary groups* Symplectic and unitary groups with $q$ odd* The proof of Theorem 3* Unitary groups with $q$ even* The proofs of Theorems A and B* References

The author extends results of McLaughlin and Kantor on overgroups of long root subgroups and long root elements in finite classical groups. In particular he determines the maximal subgroups of this form. He also determines the maximal overgroups of short root subgroups in finite classical groups and the maximal overgroups in finite orthogonal groups of c-root subgroups.