HWnotes8 - HOMEWORK 8: GRADERS NOTES AND SELECTED SOLUTIONS...

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HOMEWORK 8: GRADER’S NOTES AND SELECTED SOLUTIONS Grader’s Notes: In general when you are trying to show that two groups G and H are not isomorphic, it’s never enough to pick one particular map and show it’s not an isomorphism. For example, in showing that the rational numbers Q under addition are not isomorphic to any proper sub-group you can’t just show that the map φ ( x ) = x 2 is not an isomorphism because there could, conceivably be some other map that is an isomorphism. The proper way to show that G ±≅ H in general is usually either: (1) Find some property that you can prove / have proved is preserved under isomorphism and show one of G,H have this property while the other group does not. (2) Assume that φ G H is an isomorphism (and assume nothing more about φ ) and somehow arrive at a contradiction. The properties “one-to-one”, “onto”, and “operation preserving” are properties that functions have, it is impossible for groups to have these properties, when groups G,H are isomorphic the property that holds of the groups is that there exists a function f G H such that f , the function is one-to-one, onto, and order preserving. Chapter 6, page 135, no 28 Prove the quaternion group is not isomorphic to the dihedral group D 4 . Example Solution: Use Theorem 6.2.7 and note that the Quaternions have 1 element of order 2 while D 4 has 5 elements of order 2. There are variants on this method of proof all related to the fact that if φ is an isomorphism then ² x ² = ² φ ( x . ³ Grader’s Notes: Note that when two groups G,H are given by Cayley tables, an isomorphism φ G H need not take the n × m -th entry of the Cayley table for G to the n × m -th entry of the Cayley table for H . Chapter 6, page 136, no. 38
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This note was uploaded on 10/27/2009 for the course MATH 330 taught by Professor Staff during the Spring '08 term at Ill. Chicago.

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HWnotes8 - HOMEWORK 8: GRADERS NOTES AND SELECTED SOLUTIONS...

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