By O. T. O'Meara

This quantity, the sequel to the author's Lectures on Linear teams, is the definitive paintings at the isomorphism concept of symplectic teams over critical domain names. lately came across geometric tools that are either conceptually basic and strong of their generality are utilized to the symplectic teams for the 1st time. there's a whole description of the isomorphisms of the symplectic teams and their congruence subgroups over crucial domain names. Illustrative is the theory $\mathrm{PSp}_n(\mathfrak o)\cong\mathrm{PSp}_{n_1}(\mathfrak o_1)\Leftrightarrow n=n_1$ and $\mathfrak o\cong\mathfrak o_1$ for dimensions $\geq 4$. the recent geometric method utilized in the booklet is instrumental in extending the idea from subgroups of $\mathrm{PSp})n(n\geq6)$ the place it was once recognized to subgroups of $\mathrm{P}\Gamma\mathrm{Sp}_n(n\geq4)$ the place it's new. There are large investigations and several other new effects at the extraordinary habit of $\mathrm{P}\Gamma\mathrm{Sp}_4$ in attribute 2. the writer starts off basically from scratch (even the classical simplicity theorems for $\mathrm{PSp}_n(F)$ are proved) and the reader want be conversant in not more than a primary path in algebra.

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