Unformatted text preview: 2.7. QUOTIENT GROUPS AND HOMOMORPHISM THEOREMS 145 is the complex projective linear group; refer back to Example 2.7.11 . Let A D SL .n; C / . Then AZ D G . In fact, for any invertible matrix X , let be a complex n th root of det .X/ . then we have X D X , where X D 1 X 2 A . On the other hand, A \ Z is the group of invert- ible scalar matrices with determinant 1; such a matrix must have the form E where is an n th root of unity in C . We have G=Z D AZ=Z D A=.A \ Z/ D A= f E W is an n th root of unity g . The same holds with C replaced by R , as long as n is odd. (We need n to be odd in order to be sure of the existence of an n th root of unity in R .) If n is odd, then GL .n; R / D SL .n; R /Z . But if n is odd, the only n th root of unity in R is 1, so SL .n; R / \ Z D f E g . We see that for n odd, the projective linear group GL .n; R /=Z is isomorphic to SL .n; R /= f E g D SL .n; R / ....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08