Unformatted text preview: f e;r 2 g ; then N is normal because jr 2 j D r ± 2 D r 2 . What is G=N ? The group G=N has order 4 and is generated by two commuting elements rN and jN each of order 2. (Note that rN and jN commute because rN D r ± 1 N , and jr ± 1 D rj , so jrN D jr ± 1 N D rjN .) Hence, G=N is isomorphic to the group V of symmetries of the rectangle. Let A D f e;j g . Then AN is a four–element subgroup of G (also isomorphic to V ) and AN=N D f N;jN g Š Z 2 . On the other hand, A \ N D f e g , so A=.A \ N/ Š A Š Z 2 . Example 2.7.20. Let G D GL .n; C / , the group of n-by-n invertible com-plex matrices. Let Z be the subgroup of invertible scalar matrices. G=Z...
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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