LECTURES ON LINEA R GROUP S 3 TT(Ur\W)=TrUnnW, TT(U+ W)=-nU+-nW, dimF TTU= dimFU, TTO=0, TTV=V 1 holds for all elements U and W of P(V). In particular n carries Pl{V) onto P^Fj) , and i: is determined by its values on P l (V), i.e., it is determined by its values on lines. If g : V » V x i s geometric, then the mapping g :P(V) » P(V 1 ) obtaine d from g: pow V » po w Vx b y restriction is a projectivity o f V ont o F t . An y projectivity n:P(V) » i^Fj) whic h has the form fl^g 7 fo r such a g wil l be called a projective geometri c transforma - tion of V ont o V x . Th e bar symbol will always be used to denote the projective geometri c transformation g obtaine d from a geometric transformation g i n the above way. S o g send s the subspace U o f F , i.e. , the point U i n P(F) , t o the subspace gU o f V x . W e have gx "'gf-gi •* gt unde r composition, and g _1 =g~ l fo r inverses. In particular, composites and inverses of projective geometric transformations ar e themselves projective geometric transformations . A geometric transformation o f V is , by definition, a geometric transformation o f V on - to V. Th e se t o f geometri c transformation s o f V i s a subgroup o f th e grou p o f permuta - tions of V. I t wil l be written AL n (V) an d wil l be calle d th e genera l geometric grou p of V. B y a group o f geometric transformation s o f V w e mean an y subgrou p o f EL W (K). Th e general linear group GL n (V) consistin g o f al l invertible linea r transformation s o f V, an d the specia l linear group SL n (V) = {a E GL n (V) | det a = 1} , ar e therefor e group s o f geo- metric transformations . B y a group o f linear transformation s o f V w e mean an y subgrou p of GL n (V). B y looking a t th e determinan t homomorphis m w e obtai n GLn(V)ISLn(V)~F. A projectivity o f V is , by definition , a projectivity o f V ont o V. Th e se t o f pro- jectivities o f V i s a subgroup o f th e grou p o f permutations o f P(V) whic h wil l be called the grou p o f projectivitie s o f V. Th e bar mappin g the n provide s a homomorphism ~: *ZL n (y) -- group o f projectivitie s o f V. We sometimes use P instea d o f ~ an d pu t PX = X for th e imag e X o f a subset X o f ZL n (V) unde r P. I n particula r PGL n (V) an d PSLn(V) ar e subgroups o f th e group o f projectivities o f V called , respectively , th e projec - tive general linear grou p an d th e projectiv e specia l linea r grou p o f V. W e will se e later tha t P^Ln{V) (n 1 ) i s the entir e grou p o f projectivitie s o f V so , once thi s is done, we will be entitled t o us e thi s symbol fo r thi s group. B y a group o f projectivitie s o f V w e mean any subgrou p o f th e grou p o f projectivitie s o f V. B y a projective grou p o f linea r transfor - mations we mean an y subgrou p o f PGL n (V).
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