Hints for Homework 11

Chapter 8: 2, 4, 5, 10, 11, 12

2: You can argue by induction, but it's probably better to do this
like the case for n=2.  Go through the argument with (g_1,...,g_n) in
place of (g_1,g_2) and try to generalize.

4: The group operation in the direct product is the group operation on
each component.  Write out what it means for two elements in the
direct product to commute, then apply the Abelian property for each
component.

5: The identity element is (0,0,0).  The order of (1,0,0) is 2,
because (1,0,0) + (1,0,0) = (2,0,0), which is the identity because 
2 = 0 in Z_2.  What is the order of any non-identity element?

10: Think about the ways we know to show that two groups are not
isomorphic, for instance, by considering the possible orders of
elements.

11: Apply Theorem 8.2.  What do you know about two cyclic groups of
the same order?

12: Consider elements of the form (a,b).  If b has order 9 in Z_9,
what is the order of (a,b)?  In Chapter 4, we learned how many b have
order 9.