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College Algebra Exam Review 110

College Algebra Exam Review 110 - 120(a(b(c 2 BASIC THEORY...

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120 2. BASIC THEORY OF GROUPS (a) For A 2 GL .n; R / and b 2 R n , define the transformation T A; b W R n ! R n by T A; b . x / D A x C b . Show that the set of all such transformations forms a group G . (b) Consider the set of matrices A b 0 1 ; where A 2 GL .n; R / and b 2 R n , and where the 0 denotes a 1- by- n row of zeros. Show that this is a subgroup of GL .n C 1; R / , and that it is isomorphic to the group described in part (a). (c) Show that the map T A; b 7! A is a homomorphism from G to GL .n; R / , and that the kernel K of this homomorphism is iso- morphic to R n , considered as an abelian group under vector ad- dition. 2.4.21. Let G be an abelian group. For any integer n > 0 show that the map ' W a 7! a n is a homomorphism from G into G . Characterize the kernel of ' . Show that if n is relatively prime to the order of G , then ' is an isomorphism; hence for each element g 2 G there is a unique a 2 G such that g D a n . 2.5. Cosets and Lagrange’s Theorem Consider the subgroup H D f e; .1 2/ g S 3 . For each of the six elements
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