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4023f11ps3 - G . (c) List the elements of order 10 in G ....

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Homework #3 Due: September 14, 2011 1. Show that the multiplicative group Z / 14 is cyclic. Find all of its generators. 2. In each case determine whether G is a cyclic group. (a) G = Z / 7 (b) G = Z / 12 (c) G = Z / 16 (d) G = Z / 11 3. Find all of the generators of: (a) Z 5 (b) Z 10 (c) Z 16 (d) Z 18 4. Let G be a group and let a 2 G be an element of order 20. Compute: (a) o ( a 2 ) (b) o ( a 8 ) (c) o ( a 15 ) (d) o ( a 3 ) 5. Let a be an element in a group G such that a 15 = e . What are the possibilities for the order of a ? 6. Let G be a cyclic group of order 30 with generator a . (a) List the elements of order 2 in G . (b) List the elements of order 3 in
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Unformatted text preview: G . (c) List the elements of order 10 in G . 7. Determine (with explanation) if each of the following groups is cyclic: (a) Z 3 £ Z 9 (b) Z 3 £ Z 10 8. How many elements of order 4 does the group Z 4 £ Z 4 have? Explain why Z 4 £ Z 4 has the same number of elements of order 4 as does the group Z 400 £ Z 800 . Generalize to the case Z 4 m £ Z 4 n where m and n are arbitrary positive integers. Math 4023 1...
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This document was uploaded on 12/28/2011.

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