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By Langmuir I.
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Example text
Then XPI and XP2 are conJugate. (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.
2. But then KIT = G, and it now follows that I}(I = IG : ITIII< n ITI = IHI, a contradiction. 1 we have Hn < KnIT < IT. A-F. lQll write J('f! for the IT-associate \)\) ...... , of ]{. H. ii. If]{ is of type GL](2) I Sn in Cz(GL n(2)) with n ... " ...... ,,~ ............ " " ..... '"'' v j ..... " ..... J... 11,L,I,j, '-' /J tJ ot(q)@O~n(q). inIT with HO,i E C4 (G). :i of tensor even, product subgroups of type Sp2(q) ® SPn/2(Q) in C4 (IT). Furthermore, the last row of then ]{ nIT = Kfl x 2 ~ Sn x 2.
He Hi. H now columns 2 and 4 we give Cj and Cj, where Ho E Cj(IT) and ](0 E Cj(IT). Conditions' that ](1,](2,]{3, ](,1 are representatives for the four classes. In fact, 1':1 is Lhe under which the triple occurs appear in Column 6. Each row in the table corr Jji;';:: llnique member of [K l ] which contains HI; a similar statement holds for the other K j . to a unique triple (Ho,I{o,IT) up to conjugacy in IT, except when the term '(two)' 1::; i ::; 4, let ](j = KO,i n IT with ](O,i E C4 (G). '(four)' appears.