Hints for Homework 12
Chapter 8: 14, 15, 22
14: This question asks you to show that D_n is not isomorphic to the
direct product of Z_n + Z_2.
One approach is to count the number of elements of order 2.
Another approach is to use the fact that Z_n and Z_2 are both abelian.
22: Consider which groups are abelian to eliminate two possibilities.
Find the largest possible order of an element to eliminate a third
possibility.
Chapter 9: 1, 3, 4, 7, 12
3: See Lecture 22. We discussed this example, but you should write
out the argument more carefully than on the slide.
4: Apply the Normal Subgroup Test. Suppose you could swap the rows
and swap the columns of an element of H. Would the result always be
again in H? Find a matrix that does this swapping.
7: Choose an element a not in H. How many left (or right) cosets does
H have in G? Show that aH = Ha.
12: Find the least m>0 such that m*(14 + <8>) is the identity 0 + <8>
in Z_24/<8>.