hw5 - 5. Let G be the symmetry group of a cube (with no...

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Math 113 Homework 5, due 2/23/2012 at the beginning of section Policy: if you worked with other people on this assignment, write their names on the front of your homework. Remember that you must write up your solutions independently. 1. (a) Recall the notation for elements of the dihedral group from Homework 3. Find the left and right cosets of the subgroup H = { r 0 ,s 0 } of D 4 . Are they the same? (b) Same question for H = { r 0 ,r 2 } . Recall the group table for D 4 : r 0 r 1 r 2 r 3 s 0 s 1 s 2 s 3 r 0 r 0 r 1 r 2 r 3 s 0 s 1 s 2 s 3 r 1 r 1 r 2 r 3 r 0 s 1 s 2 s 3 s 0 r 2 r 2 r 3 r 0 r 1 s 2 s 3 s 0 s 1 r 3 r 3 r 0 r 1 r 2 s 3 s 0 s 1 s 2 s 0 s 0 s 3 s 2 s 1 r 0 r 3 r 2 r 1 s 1 s 1 s 0 s 3 s 2 r 1 r 0 r 3 r 2 s 2 s 2 s 1 s 0 s 3 r 2 r 1 r 0 r 3 s 3 s 3 s 2 s 1 s 0 r 3 r 2 r 1 r 0 2. Fraleigh section 10, exercises 34, 39, 44. 3. (a) Use the fundamental theorem of ±nitely generated abelian groups to list all abelian groups of order 16 up to isomorphism. (There are ±ve). (b) Show directly (without using the fundamental theorem) that none of these groups is isomorphic to any of the others. 4. Fraleigh section 11, exercises 6, 20, 38, 46.
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Unformatted text preview: 5. Let G be the symmetry group of a cube (with no reections allowed), i.e. the group of space rotations which preserve a cube. Show that G S 4 . Hint: a cube has four diagonals which connect opposite vertices and go through the center of the cube. Any element of G induces a permutation of the set of diagonals. It is enough to show that every permutation of the set of diagonals is realized by some element of G . (Why? What is the order of G ?) (If it helps, you can use without proof the fact that an isometry of space which xes the origin and is represented by a matrix of determinant +1 is a rotation about some axis). 6. How challenging did you nd this assignment? How long did it take?...
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