Math 416 - Abstract Algebra

Chapter 5 - Permutations

Practice  questions

The following questions are based on sample questions from the "Instructors' Solutions Manual.

1.         Find the order of the permutation (124)(2345)

2.         Give two reasons why the set of odd permutations in S_n­  is not a subgroup.

3.     a.            Write (12345) as a product of 2-cycles

b.            Write it as a product of 3-cycles

4.         In S_n, let b = (12)(123)(1234)(12345)…(123…n)

If n = 99, determine whether b is odd or even. 

5.         Let n be an integer that is 3 or larger.  How many elements of S_n send 1 to n – 2.

6.         Let b be a cycle of length at least 3 in S_n­.  Prove that b² is a cycle if and only if the length of b is odd.

7.         Prove that every non-identity permutation in S_n can be written as a product of at most n – 1 2-cycles.

8.         Let b be some fixed odd permutation in S_n­.  Prove that every odd permutation in S_n can be written as the product of b and some even permutation.

9.         Write the permutation (13)(2,4,5) as a 5x5 matrix.

10.       Show that A₈ contains an element of order 15.

11.       What is the order of each of the following permutations?

a.            (124)(357)

b.            (124)(35)

c.            (1235)(24567)

d.            (345)(245)

e.            (1234)²

12.            Determine whether the following permutations are even or odd.

a.            (135)

b.            (1356)

c.            (12)(134)(152)

13.       Prove that if a is an odd permutation, then a^-1 is an odd permutation.

14.       What is the inverse of b = (142857) ?

The test will be strongly based on these questions.

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