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- An introduction to rings, subrings, integral domains, ideals, factor rings, and ring homomorphisms
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- Fields, vector spaces, extension fields, algebraic extensions, and finite fields
- Advanced topics such as geometric constructions, Sylow Theorems, conjugacy classes, finite simple groups, and Galois theory

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**Q1:**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.See Answer**Q2:**Exercise 3 Let G be a group of order 52-72-19-23275. (a) Prove that G contains exactly one subgroup of order 49. Prove furthermore that if N <G with |N| then N is normal. (b) Prove that G/N is isomorphic to either Z19 × Z25 or Z19 × Z5 X Z Suggestion: Similar to the Exercise 2c, exept apply Proposition 3.7.1 instead of 3.7.7. 49 (c) Let Ps and P19 be Sylow 5- and 19-subgroups of G, respectively. Prove that NP5 and NP19 are both subgroups of G and that NPN X P5, and NP19 N x P19-See Answer**Q3:**Experiment 1 Tensile Testing of Rubber Apply the load (force) in the steps suggested in the table, Between steps do not release the load and try to take about the same time for eaSee Answer**Q4:**Exercise 5. FORWARD TRANSFORM For the following continuous time signals x(t), determine the Fourier transform X(jw)See Answer**Q5:**Exercise 3.6.4 Suppose that G₁, G₂ are groups, N₁ ◄G₁, N₂ ◄ G₂ are normal subgroups. Prove that N₁ × N₂ ◄ G₁ x G₂, and (G₁ × G₂)/(N₁ × N₂) ≈G₁/N₁ × G₂/N₂. Hint: Theorem 3.6.5.See Answer**Q6:**Exercise 3.7.2 Suppose G is a finite group, H, KG are normal subgroups, ged(|H|, |K|) = 1, and |G| = |H||K|. Prove that G H x K.See Answer**Q7:**Exercise 3.7.6 Prove that for all n < 2, we have Sn = An * Z2, where Sn is the symmetric group and An theSee Answer**Q8:**Cancellation laws: use only rules from 'basics' to prove cancellation laws for addition and multiplication for natural numbers. That is, prove for natural numbers m, n, k, i) if m + k = n + k then m = n, ii) if mk nk then m = n, iii) if m + k <n+k then m <n, iv) if mk < nk then m <n, v) if m + k ≤n + k then m≤n, vi) if mk ≤ nk then m≤ n.See Answer**Q9:**a) Calculate possible remainders of a prime p > 3 when divided by 6 b) Prove for any prime p > 3, the number p² + 1 can't be prime.See Answer**Q10:**1. Cancellation laws: use only rules from 'basics' to prove cancellation laws for addition and multiplication for natural numbers. That is, prove for natural numbers m, n, k, i) if m + k = n + k then m = n, ii) if mk = nk then m = n,See Answer**Q11:**5. a) Calculate possible remainders of a prime p > 3 when divided by 6. 5. b) Prove for any prime p > 3, the number p² + 1 can't be prime.See Answer**Q12:**If a family borrows $12,825 for an addition to their home, and the loan is to be paid off in monthly payments over a period of 5 years, how much should each payment be? (Interest has been included in the total amount borrowed.) Round your answer to the nearest cent, if necessary. Each payment should beSee Answer**Q13:**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 XSee Answer**Q14:**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 5See Answer**Q15:**A fraternity charged $2.00 admission for dudes and $1.00 admission for ladies to their finals week bash. The fraternity made $55.00 and sold 45 tickets. How many ladies attended the party? There were ladies at the party.See Answer**Q16:**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 5See Answer**Q17:**When a retired police officer passes away, he leaves $300,000 to be divided among his three children and six grandchildren. The will specifies that each child is to get twice as much as each grandchild. How much does each get? Each child gets $ and each grandchild gets XSee Answer**Q18:**If 116 is added to a number, it will be 5 times as large as it was originally. Find the number. The number is XSee Answer**Q19:**Use the information shown in the graph. The graph represents a survey of 1325 office workers and shows the percent of people who indicated what time of day is most productive for them. Most Productive Time of Day After office hours 9% Last few office hours 14% First few working hours 31% Before office hours 25% Late morning/early afternoon 21% Estimate the number of people who feel they are most productive outside normal office hours. Round your answer to the nearest integer. An estimate for the number of people who feel they are most productive outside normal office hours is XSee Answer**Q20:**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 XSee Answer

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