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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Consider a regular plane IT in V, fix a line L o in IT, and let TLa be a transvection in SPn (V) which is projectively in Li and whose residual line is Lo. For each line X that falls in IT but is distinct from Lo let T X be defined in the same way. 3, and they are projectively in Li. , it is equal to q. 4. Therefore card Li > qn- I ( :: =;). 2. D. 2. 1. Suppose n > 4, n l > 4. Let T be a transvection in SPn (~/) that is projectively in Li, and let k l be an element of fSPn,( VI) "It'ith Ai = k ,.

We then have, for any u E U, w E W, + w, a(u + w» = q(u + w, u + aw) = O. Q. E. D. Ll3. LlI is true. PROOF. (I) First suppose the given a is not hyperbolic. If we can find a transvection r in SPn (V) such that res m < res a with ra either nonhyperbolic 26 O. T. O'MEARA or I v' then by a successive application of this fact, we will be through. So our purpose in step (I) is to establish this fact. (la) We can suppose that peR. For suppose this case is already known and consider a a with P g R. Then P is not totally degenerate.

D. Ta,I' o So G contains all transvections in SPn( V). 6. If n ;;;. 4, then DSPn (SPiF2): DSP4(F 2») = 2. PROOF. Put 'T = 'Ta . 1 for some a in such that Fa + F}:;a is a plane with V. 5. 5 plus the well-known behavior of 6 6 , QED. 7. If n ;;;. 4, then D PSPn = PSPn but for the following exception: (PSp4(F2): DPSP4(F2») = 2. 4. 1. THEOREM. The group PSPn(F) is simple for any natural number n ;;;. 4 and any field F but for the group PSpiF2) which is not simple. PROOF. 7. So let us assume that n ;;;.