Description

This volume, the sequel to the author's Lectures on Linear Groups, is the definitive work on the isomorphism theory of symplectic groups over integral domains. Recently discovered geometric methods which are both conceptually simple and powerful in their generality are applied to the symplectic groups for the first time. There is a complete description of the isomorphisms of the symplectic groups and their congruence subgroups over integral domains. Illustrative is the theorem $\mathrm{PSp}_n(\mathfrak o)\cong\mathrm{PSp}_{n_1}(\mathfrak o_1)\Leftrightarrow n=n_1$ and $\mathfrak o\cong\mathfrak o_1$ for dimensions $\geq 4$. The new geometric approach used in the book is instrumental in extending the theory from subgroups of $\mathrm{PSp})n(n\geq6)$ where it was known to subgroups of $\mathrm{P}\Gamma\mathrm{Sp}_n(n\geq4)$ where it is new. There are extensive investigations and several new results on the exceptional behavior of $\mathrm{P}\Gamma\mathrm{Sp}_4$ in characteristic 2. The author starts essentially from scratch (even the classical simplicity theorems for $\mathrm{PSp}_n(F)$ are proved) and the reader need be familiar with no more than a first course in algebra.