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>.