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Some Questions on Cyclic Groups (9/27)

Some Questions on Cyclic Groups (9/27). Is Q (under addition) cyclic? A. Yes B. No Does Q possess cyclic subgroups? A. Yes B. No Is Q * (under multiplication) cyclic? A. Yes B. No Does Q * possess nontrivial finite cyclic subgroups: A. Yes B. No

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Some Questions on Cyclic Groups (9/27)

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  1. Some Questions on Cyclic Groups (9/27) • Is Q (under addition)cyclic? A. Yes B. No • Does Q possess cyclic subgroups? A. Yes B. No • Is Q* (under multiplication) cyclic? A. Yes B. No • Does Q* possess nontrivial finite cyclic subgroups: • A. Yes B. No • Is Dncyclic (n 3)? A. Yes B. No • Does Dn possess cyclic subgroups? A. Yes B. No • Does every group G possess cyclic subgroups? • A. Yes B. No • Does every group G possess nontrivial, proper cyclic subgroups? A. Yes B. No • Is U(15) cyclic? A. Yes B. No • Is U(14) cyclic? A. Yes B. No

  2. More Questions • What is the smallest generator of 20 in Z35? • A. 1 B. 5 C. 7 D. 20 • What is the order of 20 in Z35? • A. 1. B. 5 C. 7 D. 35 • If a  G and |a| = 32, what is the smallest exponent k on a such that akgenerates a28? • A. 2 B. 4 C. 8 D. 16 E. 28 • What is the order of a28 inside a (a as in last question)? • A. 2 B. 4 C. 8 D. 16 E. 32 • What is a generator of 8, 10, 12 in Z? • A. 1 B. 2 C. 4 D. 12

  3. More Questions • Suppose |a| =  (a an element of a group G).How any generators does a have? • A. 1 B. 2 C. Infinitely many • How many generators does Z52 have? • A. 1 B. 2 C. 12 D. 24 E. 36 • How many generators does Z53 have? • A. 2 B. 21 C. 40 D. 52 E. 53 • U(34) is cyclic. How many generators does it have? • A. 2 B. 4 C. 8 D. 16 E. 33 • How many generators does 3 in U(20) have? • A. 1 B. 2 C. 4 D. 8 E. 19

  4. And Yet a Few More • How many non-trivial proper subgroups does D4 have? • A. 1 B. 4 C. 6 D. 8 E. 12 • How many non-trivial proper subgroups does Z8have? • A. 0 B. 1 C. 2 D. 4 E. 8 • How many non-trivial proper subgroups does Z24have? • A. 0 B. 1 C. 4 D. 6 E. 12 • How many non-trivial proper subgroups does Z23have? • A. 0 B. 1 C. 2 D. 6 E. 10 • How many non-trivial proper subgroups does Zp have (p a prime number)? • A. 0 B. 1 C. 2 D. p – 1 E. p • How many non-trivial proper subgroups does Zp^2have? • A. 0 B. 1 C. 2 D. p2 – p E. p2

  5. Assignment for Monday • Take a fresh sheet of paper, copy the statement of Theorem 4.1, then close the book and your notebook and write a careful proof of the theorem on your own. • On page 88, do Exercises 15, 16, 17, 18, 19, 22, 23.

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