Hints for Homework 7

4.60: Apply theorems 4.2 and 4.3.  Rewrite <x^r> and <x^s> using a
gcd, then consider a divisibility condition.

5.13: Prove that A_n is a subgroup of S_n using the Two-step subgroup
test.  Exercise 5.12 will help.  To show closure, consider what
happens when you write two even permutations each as a product of
2-cycles.  How many terms are in the product when you compose them?

5.21: What happens when you compose two odd permutations?  If you
write them as a product of 2-cycles, will there be an odd or even
number of terms in the product?

6.1, 6.3: Be sure to specify the map and show the four properties
of an isomorphism.

6.2: I didn't cover the material on automorphisms well enough, so I
would like to delay this problem until Homework 8.  You don't have to
submit it with this set.

6.5: A straightforward way to show that an isomorphism preserves the
group operation is to write down a multiplication table for both
groups and show that an isomorphism amounts to a relabelling of
the tables.  When you write out the tables, you can see what map(s)
would make an isomorphism.