Mathematics 581   Problem Set 1   Due Monday, April 8

Textbook Abstract Algebra, by W. E. Deskins. Dover, 1992. ISBN: 0-486-68888-7 (pbk.) The library copy is on closed reserve in the Sci/Eng Library.

1. (a) Write as a product of disjoint cycles the permutation given in 2-row notation by

1 2 3 4 5 6 7 8 9

3 7 9 4 8 2 6 5 1

(b) Write as a product of disjoint cycles the permutation

(1 2 7)(2 8 9)(9 7 3 4)(4 7)

(c) Write in 2-row notation the 4-cycle (1 3 2 4)

2. (a) Show that the inverse of the permutation  (1 2 7) is  (7 2 1).

(b) Explain why the inverse of any n-cycle can be found by simply writing the entries in the cycle in reverse order.

(c) Is part (b) true for products of cycles? Explain. Hint: Whatever you think is going to be your answer, test it on the permutation  (1 2)(2 3 4).

3. Textbook, p. 197, #1:  List all permutations of the set  {a, b}  and using composition of functions as the multiplication, write out the table of this group of permutations..

4. Textbook, p. 197, #4. (Hint: In Example 3 every group element is its own inverse.  Is the same true in Example 5?)

5. Prove that if  a  is an element of a group  G  and  a2 = a, then  a  is the identity element of  G.  (Hint: If e  is the identity element, then  a2 = a = ae.  Now multiply on the left by the inverse of  a.)

6. (a) Let  a  be an element of a group G. Let  a'  be the inverse of a.  Prove that the inverse of  an  is(a')n, for any positive integer  n.

(b) Do you think that part (a) is true if  n  is negative?  Explain.