By Harde K. A.

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**Example text**

Ii) Any G-invariant subspace of V is a sum of some of tbe Vi, Proof. (i) We use induction on t, and we define V;- 55 Basic Properties of the Classical Groups as G-modules. Proof. Assume that (Xp1)g = XP2 for some 9 E GL(V, F). Then there is a map B : X --+ X which satisfies (xpJ)g = (X())P2. Clearly () is a well-defined automorphism of X since thepi are faithful. Thus PI is equivalent to 8P2, so PI and P2 are quasiequivalent. The converse is just as easy, and the details are left to the reader.

In case L there is also an occurrence of type. 1. More generally, all the types will be described explicitly throughout Chapter 4. With little doubt, the best way for the reader to become familiar with our conventions concerning types is by way of examples. 1 [As 1 , Theorem BTl The group A acts transitively on the groups in C(r) of a given type T. The group r also acts transitively, except for the fact that in case L with has two orbits on groups of type Pi when i -:f. 11 - i. 11. A-F, and vve now describe briefly the other columns.

Im(n)\ = ex. H type of Ho 1(0 ~ conditions Sp2(q) 1St E C7 SP2t( q) ~t(q) q even, q 2: 4, t 2: 4 11 2: 13 two classes Sp2(q) I St E C7 ~it+2(q) ~t(q) q even, q 2: 4, todd, t 2: 5 two classes Sp4(q) I St E C7 SP4t(q) L C1 L± C7 L Cs N -E= I :;. 0 "'IJ ::... 1. i holds. 0 Introduction ~n this. c~apt~r we describe the natural collections Cj of subgroups of classical groups, m addltlOn gIve complete information about their structure and conjugacy. 15)). 1 that the members of Care de£ned in terms of the members of C(r) (see (3.