College Algebra Exam Review 102

College Algebra Exam Review 102 - trix is nonzero and that...

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112 2. BASIC THEORY OF GROUPS .1432/ , '.a/ D .14/.23/ , and '.c/ D .24/ . Now you can compute that '.a/'.r/ D .14/.23/.1432/ D .24/ D '.c/ D '.ar/: You are asked in Exercise 2.4.1 to complete the tabulation of the map ' from the symmetry group of the square into S 4 and to verify the homo- morphism property. Note that all of this is a formalization of the computation by pictures that was done in Section 1.3 . 1 2 1 2 Figure 2.4.2. Labeling the diagonals of the square. Example 2.4.4. There are other sets of geometric objects associated with the square that are permuted by symmetries of the square: the set of edges, the set of diagonals, the set of pairs of opposite edges. Let’s consider the diagonals (Figure 2.4.2 ). Numbering the diagonals gives a homomorphism from the group of symmetries of the square into S 2 . You can compute, for example, that .r/ D .a/ D .12/ , while .c/ D e . You are asked in Exercise 2.4.2 to complete the tabulation of the map and to verify its homomorphism property. Example 2.4.5. It is well known that the determinant of an invertible ma-
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Unformatted text preview: trix is nonzero, and that the determinant satisfies the identity det .AB/ D det .A/ det .B/ . Therefore, det W GL .n; R / ! R ± is a homomorphism from the group of invertible matrices to the group of nonzero real numbers under multiplication. Example 2.4.6. Recall that a linear transformation T W R n ! R n has the property that T. a C b / D T. a / C T. b / . Thus T is a group homomorphism from the additive group R n to itself. More concretely, for any n-by-n matrix M , we have M. a C b / D M a C M b . Thus multiplication by M is a group homomorphism from the additive group R n to itself. Example 2.4.7. Let G be any group and a and element of G . The the map from Z to G given by k 7! a k is a group homomorphism. This is equivalent to the statement that a k C ` D a k a ` for all integers k and ` . The image of this homomorphism is the cyclic subgroup of G generated by g ....
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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.

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