10Ma5aHw2Sol

10Ma5aHw2Sol - Ma 5 a HOMEWORK ASSIGNMENT 2 SOLUTIONS FALL...

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Unformatted text preview: Ma 5 a HOMEWORK ASSIGNMENT 2 SOLUTIONS FALL 2010 The exercises are taken from the text, Abstract Algebra (third edition) by Dum- mitt and Foote. Page 22, 25 . Take a, b G . Then ab G , and ( ab )( ba ) = ab 2 a = 1 = ( ab )( ab ). Multiplying both sides of ( ab )( ba ) = ( ab )( ab ) on the left by ( ab )- 1 , we get ba = ab . Therefore G is abelian. Page 28, 10 . We claim that the group of rigid motions of a cube are obtained from rotating the cube about one of the 3 types of axis 1) connecting the centers of 2 opposite faces, 2) connecting the midpoints of 2 opposite edges, and 3) connecting a pair of 2 opposite vertices. For the first type, rotation about a given axis has order 4. There are 3 such axes, so we get 9 distinct rigid motions, not counting the identity. For the second type, each rotation has order 2. With 6 such axes, we get 6 distinct elements in the group not counting the identity. For the last type, each rotation has order 3. 4 such axes give 8 distinct elements in the group not counting the identity. In total, we have 9 + 6 + 8 + 1(for the identity)= 24. Letcounting the identity....
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10Ma5aHw2Sol - Ma 5 a HOMEWORK ASSIGNMENT 2 SOLUTIONS FALL...

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