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Group Theory Questions and Assignments: Exploring Cyclic, Dihedral, and Permutation Groups

Delve into questions on cyclic, dihedral, and permutation groups to deepen your understanding. Explore the largest order of elements in various groups, proper subgroups, and more. Prepare for your test with this comprehensive assignment.

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Group Theory Questions and Assignments: Exploring Cyclic, Dihedral, and Permutation Groups

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  1. Some Questions on Cyclic, Dihedral and Permutation Groups (10/7) • What is the largest order of an element in Z10? • A. 2 B. 5 C. 10 D. 4 E. 8 • What is the largest order of an element in D10? • A. 2 B. 5 C. 10 D. 20 E. 12 • What is the largest order of an element in S10? • A. 10 B. 15 C. 21 D. 30 E. 60 • What is the largest order of an element in A10? • A. 10 B. 15 C. 21 D. 30 E. 60 • S10 has 3628800 elements. What is the largest order of an element in the dihedral group of that same order? • A. 10 B. 800 C. 1814400 D. 3628800 E.

  2. More Questions • How many non-trivial proper subgroups does D3 have? • A. 2 B. 3 C. 4 D. 5 E. 6 • How many non-trivial proper subgroups does S3 have? • A. 2 B. 3 C. 4 D. 5 E. 6 • How many non-trivial proper subgroups does Z20have? • A. 0 B. 3 C. 4 D. 6 E. 10 • How many non-trivial proper subgroups does Z19have? • A. 0 B. 2 C. 3 D.5 E. 6 • How many non-trivial proper subgroups does D7have? • A. 2 B. 7 C. 8 D. 10 E. 14 • How many non-trivial proper subgroups does D8have? • A. 6 B. 8 C. 12 D. 15 E. 18 (Toughy!!)

  3. And some more... • How many subgroups of order 8 does Z14 have? • A. 0 B. 1 C. 2 D. 6 E. 7 • How many subgroups of order 7 does Z14 have? • A. 0 B. 1 C. 2 D. 6 E. 7 • How many elements of order 7 does Z14 have? • A. 0 B. 1 C. 2 D. 6 E. 7 • How many elements of order 6 does S5 have? • A. 0 B. 3 C. 10 D. 20 E. 40 • How many elements of order 6 does A5have? • A. 0 B. 3 C. 10 D. 20 E. 40 • How many subgroups of order 6 does S5 have? • A. 0 B. 3 C. 10 D. 20 E. 40 (Toughy also!)

  4. Assignment for Wednesday • Review everything. Prepare questions. Start thinking about and/or creating your reference sheet. • Test #1 is on Friday!

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