Symmetry And Group

## Symplectic Groups by O. T. O'Meara

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By O. T. O'Meara

This quantity, the sequel to the author's Lectures on Linear teams, is the definitive paintings at the isomorphism conception of symplectic teams over necessary domain names. lately stumbled on geometric equipment that are either conceptually basic and robust of their generality are utilized to the symplectic teams for the 1st time. there's a entire description of the isomorphisms of the symplectic teams and their congruence subgroups over indispensable domain names. Illustrative is the concept $\mathrm{PSp}_n(\mathfrak o)\cong\mathrm{PSp}_{n_1}(\mathfrak o_1)\Leftrightarrow n=n_1$ and $\mathfrak o\cong\mathfrak o_1$ for dimensions $\geq 4$. the hot geometric method utilized in the booklet is instrumental in extending the speculation 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 truly is new. There are large investigations and a number of other new effects at the extraordinary habit of $\mathrm{P}\Gamma\mathrm{Sp}_4$ in attribute 2. the writer begins basically from scratch (even the classical simplicity theorems for $\mathrm{PSp}_n(F)$ are proved) and the reader desire be accustomed to not more than a primary direction in algebra.

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Such that Fa + F}:;a is a regular plane. (3) We can also show that we have a E V, }:; E G such that Fa + F}:;a is a degenerate plane. Pick E G with =I' ± 1v' Then there is an a in V such that Fa =I' Fa. If q(a, a) = 0 we are done, so assume q(a, a) =I' O. Pick b E V - (Fa + Fa) with q(b, a) =0 and q(a, b) = q(a, a) =I' O. O. 42 T. O'MEARA By Witt's Theorem there is a T in SPn( V) with Ta =

Oa» will do the job. The fixed space of this Ta -I is equal to P + Fa (which :J P). PROOF. First let us suppose that we have a T with res 'Ta < res a. Let L ~ H be the residual and fixed spaces of T. Of course H = L *. Put a l = 'Ta. Then the equation a = T -Ial implies that L g R I' hence H ;Q PI' hence there is an a in V with ala = a and Ta =1= a. Then q(a, aa) = q(a, T-laJa) = q(a. 1. Conversely, let there be an a in V with q(a, aa) =1= O. Put T = -I T, . 3. Hence 'Taa = a. Now the residual space of Ta is contained in F(aa - a) + R which is R since aa - a E R; hence the fixed space of Ta contains P; but a is in the fixed space of Ta though not in the fixed space P of a; hence the fixed space of Ta contains P + Fa which strictly contains P.

Then there is an X in V with (JX =I=- x. Let C and C be the two configurations containing x. Then (JX does not belong to one of them, say (JX fl C. So (JC =I=- C. So =I=- I. In other words, the kernel is trivial and we have an injective homomorphism Spi V) >- CS 6 · But Spi V) has 6! 2. So Spi V) is isomorphic to CS 6 , as required. D. 2. Centers Note that PSPn(V) is not commutative. 8. So SPn( V) is not commutative too. 1. PSPn( V) is centerless and cen SPn( V) = (± 1v). 38 O. T. O'MEARA PROOF.