College Algebra Exam Review 147

# College Algebra Exam Review 147 - .˛ a ± 1.n ± 1;a ± 1...

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3.2. SEMIDIRECT PRODUCTS 157 Remark 3.2.4. The direct product is a special case of the semidirect prod- uct, with the homomorphism ˛ trivial, ˛.a/ D id N for all a 2 A . Corollary 3.2.5. Suppose G is a group, N and A are subgroups with N normal, G D NA D AN , and A \ N D f e g . Then there is a homo- morphism ˛ W A ! Aut .N/ such that G is isomorphic to the semidirect product N Ì ˛ A . Proof. We have a homomorphism ˛ from A into Aut .N/ given by ˛.a/.n/ D ana ± 1 . Since G D NA and N \ A D f e g , every element g 2 G can be written in exactly one way as a product g D na , with n 2 N and a 2 A . Furthermore, .n 1 a 1 /.n 2 a 2 / D Œn 1 .a 1 n 2 a ± 1 1 /ŁŒa 1 a 2 Ł D Œn 1 ˛.a 1 /.n 2 /ŁŒa 1 a 2 Ł . Therefore, the map .n;a/ 7! na is an isomor- phism from N Ì ˛ A to G . n Example 3.2.6. Z 7 has an automorphism ' of order 3 , '.ŒxŁ/ D Œ2xŁ ; this gives a homomorphism ˛ W Z 3 ! Aut . Z 7 / , deﬁned by ˛.ŒkŁ/ D ' k . The semidirect product Z 7 Ì ˛ Z 3 is a nonabelian group of order 21. This group is generated by two elements a and b satisfying the relations a 7 D b 3 D e , and bab ± 1 D a 2 . Exercises 3.2 3.2.1. Complete the proof that N Ì ˛ A (as deﬁned in the text) is a group by verifying that
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Unformatted text preview: .˛ a ± 1 .n ± 1 /;a ± 1 / is the inverse of .n;a/ . 3.2.2. Show that j W ŒxŁ 7! Œ ± xŁ deﬁnes an order 2 automorphism Z n . Conclude that ˛ W Œ1Ł 2 7! j determines a homomorphism of Z 2 into Aut . Z n / . Prove that Z n Ì ˛ Z 2 is isomorphic to D n . 3.2.3. Show that the afﬁne group Aff .n/ is isomorphic to a semidirect product of GL .n; R / and the additive group R n . 3.2.4. Show that the permutation group S n is a semidirect product of Z 2 and the group of even permutations A n . 3.2.5. Consider the set G of n-by-n matrices with entries in f 0; ˙ 1 g that have exactly one nonzero entry in each row and column. These are called signed permutation matrices. Show that G is a group, and that G is a semidirect product of S n and the group of diagonal matrices with entries...
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