College Algebra Exam Review 134

College Algebra Exam Review 134 - f e;r 2 g ; then N is...

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144 2. BASIC THEORY OF GROUPS Proposition 2.7.18. (Diamond Isomorphism Theorem) Let ' W G ±! G be a surjective homomorphism with kernel N . Let A be a subgroup of G . Then (a) ' ± 1 .'.A// D AN D f an W a 2 A and n 2 N g , (b) AN is a subgroup of G containing N . (c) AN=N Š '.A/ Š A=.A \ N/ . We call this the diamond isomorphism theorem because of the follow- ing diagram of subgroups: AN ± ± ± @ @ @ A N @ @ @ ± ± ± A \ N Proof. Let x 2 G . Then x 2 ' ± 1 .'.A// , there exists a 2 A such that '.x/ D '.a/ , there exists a 2 A such that x 2 aN , x 2 AN: Thus, AN D ' ± 1 .'.A// , which, by Proposition 2.7.12 , is a subgroup of G containing N . Now applying Theorem 2.7.6 to the restriction of ' to AN gives the isomorphism AN=N Š '.AN/ D '.A/ . On the other hand, applying the theorem to the restriction of ' to A gives A=.A \ N/ Š '.A/ . n Example 2.7.19. Let G be the symmetry group of the square, which is generated by elements r and j satisfying r 4 D e D j 2 and jrj D r ± 1 . Let N be the subgroup
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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 fourelement subgroup of G (also isomorphic to V ) and AN=N D f N;jN g Z 2 . On the other hand, A \ N D f e g , so A=.A \ N/ A 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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