17. Prove: Um is closed under multiplication; i.e., the product of any two elements of

Use the line graph shown. The graph shows annual cigarette consumption (in billions) for the United States for the years 1900 to 2010. Cigarette Consumption in the United States Billions of cigarettes 700 600 500 400 300 200 100 0 1900 1920 1940 1960 1980 Year 2000 The year(s) in which 400 billion cigarettes were smoked: Estimate the year or years in which 400 billion cigarettes were smoked. If there is more than one answer, separate them with commas. DO 2010 X

5. (14 points) Suppose that : G→ H is a group homomorphism, K <H is a subgroup, and ¹(K) = {g €G | (g) € K}. Prove that -¹(K) is a subgroup of G. 6. (14 points) Suppose A, B4G are normal subgroups such that AnB = {e}. Prove that if a EA and be B, then ab = ba. IMPORTANT: You should not simply quote the Proposition from class stating that AB Ax B. Instead, you should supply the first part of that proof of that proposition, stating that the elements of A commute with the elements of B.

7. Let o = (15)(26 9 7 8) (34), 7= (1 3 5)(29) € Sg and G=(0,7) < Sg. (a) (7 points) Find the composition a OT, expressed in disjoint cycle notation. JOT= (b) (7 points) Is a even or odd? Circle the correct answer. EVEN ODD (c) (7 points) Compute the order of a. |o|= (d) (7 points) Considering the action of G on {1,2,3,4,5,6,7,8,9} coming from the inclusion into S9, list the elements of the orbit of G. G.1= (e) (7 points) With the action of G in part (d), we have stabc(1) has 360 elements (you do not need to prove this). What is the order of G? Hint: Use your answer to part (d), even if you couldn't find G.1. |G|=

8. Let G be a nonabelian group of order 253 =23-11, let P <G be a Sylow 23-subgroup and Q<G a Sylow 11-subgroup. (a) (7 points) What are the orders of P and Q? |P| = |2|= (b) (14 points) How many distinct conjugates of P and Q are there in G? Justify your answer (no credit without justification). 123 = the number of conjugates of P 711 = the number of conjugates of Q= (c) (7 points) Prove that G is isomorphic to a semidirect product P x Q.

At the dog park, there are several dogs with their owners. Counting heads, there are 19, counting legs, there are 60. How many dogs and owners are there? There are dogs and owners. X 5

Exercise 4.3.4 Let p≥ 2 be a prime and prove that any group of order p² is isomorphic to either Zp² or Zpx Zp. Hint: If there exists an element of order p², then GZp², so assume that this is not the case and deduce that every nonidentity element of G has order p. Then prove that for any two elements g, h E G, either (g) = (h) or (g) n(h) = {e}. Finally, prove that there are two nonidentity elements g, h E G so that (g) n(h) = {e}, and apply Proposition 3.7.1.

3. (14 points) Prove that (Z,+), the integers with addition, forms a group (you may assume "standard" facts about addition of integers). 4. (14 points) Draw the subgroup lattice of Z100-

17. Prove: Um is closed under multiplication; i.e., the product of any two elements of Um is still in Um. (This exercise is referenced in Example 8.6.) Note: Your proof may not use Theorem 4.19. But Exercise #16 above should help!

A physical therapy facility is building a new pool that is 60 feet long and 6 feet deep. They have ordered enough tile for a 220-foot-long border around the edge. How wide should the pool be to ensure that all tiles are used? The pool should be feet wide. X 5

Kam's monthly budget includes $259 for food, $145 for gasoline, and $160 for utilities. If he earns $1615 per month after taxes, how much money is left for other expenses? The amount of money left for other expenses is X