4 O . T . O'MEAR A 1.2. Radiation s For an y nonzer o a i n F defin e th e linear transformatio n r a b y rax = ax V x E V. Thus r a i s in GL n (V). An y a i n GL n whic h has the for m o = ra fo r som e suc h a will be called a radiation o f K Th e set o f radiation s o f V i s a normal subgroup of GL n (V) which will be written RL n (V). Th e isomorphism /?Z, W * F i s obvious. 1.2.1. Le r o be any element of GL n (V). Then o is in RL n (V) if and only if oL = L for all lines L in V. In particular ker (P\GL n ) = RLn, ke r (P\SL n ) = SLn n /tt «?!„ ^ CI„//?L„ , iSZ, - SLjSL n n *£„ . PROOF. Fi x z i n K . Ther e i s then a | 3 i n F suc h tha t oz = |3z. W e have t o prov e that ox = j3x fo r a typical x i n K B y hypothesis, ox = ax fo r som e a i n F . I f x is in Fz , the n x ha s the for m x = Xz, s o ax = o(\z) = X(az) = Xj3z = |3x. If x i s not i n Fz t the n ax 4 - 0z = a(x + z ) = 7(x + z) , so a = 7 = |3 b y th e independenc e o f x an d z . Q . E. D. 1.2.2. PSL n (V) is a normal subgroup of PGL n (V) with PGL n /PSLn ^ F/F n . PROOF. Normalit y i s clear. Nex t verif y tha t th e kerne l o f th e composit e homomor - phism GLn - * PGL n -^PGLJPSL n n p n n a t nl n is the grou p G = {o E GL n | det a E F"} . The n verify tha t th e kernel of th e composit e homomorphism GLn-*F-+ F/F n n de t na t ' is also G. Finally , PGLjPSLn = GLJG = F/Fn. Q.E.D . 1.3. Residue s Consider o i n GL n (V). W e define th e residua l spac e R, th e fixe d spac e P, an d th e residue re s a, o f a b y th e equation s R = (o- \ V )V, P=ker(a - \ v \ res a = dim R. The subspace s / ? an d P o f F ar e called th e space s of a . W e have
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