CyclicGroupProblems

# CyclicGroupProblems - Quiz 1 Practice Problems: Cyclic...

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Quiz 1 Practice Problems: Cyclic Groups Math 332, Spring 2010 These are not to be handed in. The quiz will be on Tuesday. 1. Find all generators of Z 6 , Z 8 , and Z 20 . 2. List all elements of the subgroup h 30 i in Z 80 . 3. If | a | = 60, what is the order of a 24 ? 4. How many subgroups does Z 20 have? List the possible generators for each subgroup. 5. Suppose that | a | = 24. Find a generator for h a 21 i ∩ h a 10 i . In general, what is a generator for the subgroup h a m i ∩ h a n i ? 6. In Z 60 , list all generators for the subgroup of order 12. 7. Let G be a group and let a be an element of G . a. If a 12 = e , what can we say about the order of a ? b. If a m = e , what can we say about the order of a ? 8. List all elements of order 8 in Z 8000000 . How do you know your list is complete? 9. Determine the subgroup lattice for Z p 2 q , where p and q are distinct primes. 10. Determine the subgroup lattice for Z p n , where p is a prime and n is some positive integer. 11.

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CyclicGroupProblems - Quiz 1 Practice Problems: Cyclic...

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