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 a^{2} = a, then
a is the identity element of G. (Hint: If e is
the identity element, then a^{2} = 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 a^{n}^{ }is(a')^{n},
for any positive integer n.

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