Published on: **Mar 3, 2016**

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- 1. Math E-301: Homework 3 Due 9/28/09 Reading: To help you start thinking about congruences, please read Friedberg, pp. 114-120. Note: There’s no reading this week about the group theory we’ve been doing. Please don’t be concerned if a lot of this is very new to you - we won’t be delving any more into it than what we’ve just done, and it shouldn’t be necessary for you to try to ﬁnd a textbook covering this material. 1. In class, we proved the cancellation property for a group with operation #: If A#B = A#C then B = C. This is true whether or not the group is abelian. (Recall that “abelian” is the same as “com- mutative”). But we also left “open” the question of whether we could cancel in the following way If A#B = C#A then B = C when the group G was non-abelian. To settle this question, let’s look at the non-abelian group D6 of the symmetry moves of an equilateral triangle, {1, R1 , R2 , F1 , F2 , F3 }. Find at least two examples where this version of the cancellation law fails in D6 . (Please be clear about how you are deﬁning your rotations and ﬂips- for example, counterclockwise or clockwise, ﬂip across axis through lower left vertex, etc). 2. Just as we did for the moves of a triangle, create a group table for the symmetries of a square. This is D8 , known as the dihedral group of the square. (Please be clear about how you are deﬁning your rotations and ﬂips- for example, counterclockwise or clockwise, ﬂip across axis through lower left vertex, etc). 3. In class, we ended with the Hershey’s Chocolate Bar game: Start with a bar, 2 squares by 6. At each turn, a player picks up any of the pieces on the table and breaks it completely along any of the existing rows or columns of the piece of bar. A player wins the game if she or he makes the last break (i.e. if after his/her break, all of the individual chocolate squares have been broken oﬀ). Should you go ﬁrst or second in order to win this game? Then explain how to win at the game if the starting bar is in the shape of an m by n rectangle. Be sure to explain why your strategy works. 4. Suppose we modify the Hershey’s Chocolate Bar game to allow three people to play. If the bar starts as a 2 by 6 rectangle, should you go ﬁrst, second or third to win? If the starting bar is in the shape of an m by n rectangle, what is the general strategy for winning in this case? Be sure to explain your answer. 5. We “know” −1 × −1 = 1 in the integers, but why? (a) Give an argument for why −1 × −1 = 1 in Z that would be appropriate to convince a student (elementary, middle school, or above). (b) What about in any ring R? Does −1 × −1 = 1? Here 1 stands for the multiplicative identity of R, and “-1” means the additive inverse of 1. Using just the properties of a ring, try to prove this. (Recall that in class, we proved that in any ring R with 0 as the additive identity, 0 × A = 0 for every A. We did this by examining (1 + 0)A and using the distributive property. Try a similar strategy). 6. Consider the set of numbers {1, −1, i, −i} where i is the square root of −1. Show this set is group under the operation of multiplication and create a group table for it. Which order four group is it equivalent to, Z4 or the Klein Four group1 ? 1 The Klein Four group is also an a cappella mathematical singing group. If you care for some uniquely mathematical humor, you can ﬁnd their ”hit” song ”Finite Simple Group (of Order Two)” on the web.
- 2. 7. Let’s take the set of numbers {1, 2, 3, 4} and make a group with the operation of multiplication modulo 5: multiply two numbers, divide by 5 and keep the remainder as the result. (For example, 2 times 4 equals 8, which when divided by 5 has remainder 3. So 2 ×5 4 = 3.) (a) Create a group table for this set of 4 elements, and check that it is in fact a group. (When you check commutativity and associativity, think about what you know about Z). (b) Which group of order four is it? There are only 2 such groups, Z4 and the Klein Four Group (also know as V for Vierergruppe). 8. Now consider Z5 = {0, 1, 2, 3, 4} with the operations of addition and multiplication modulo 5. (a) Write out the addition table for the group (Z5 , +5 ). (b) Is Z5 a commutative ring or a ﬁeld? (You may use and reference your results from the previous problem). 9. Now let’s think about what happens with Z6 with addition and multiplication modulo 6. (Again, arithmetic in Z6 is “clock arithmetic”, in that we just keep track of remainders after dividing by 6. So for example, 2×6 5 = 4 since 10 divided by 6 has remainder 4.) Is (Z6 , +6 , ×6 a commutative ring or a ﬁeld? Be sure to write out the tables for addition and for multiplication in Z6 as part of the write-up for this problem. 10. Finally, do the same for two more possible number systems: Z7 and Z8 . For each, write out the tables for both addition and multiplication modulo 7 (or 8). Then determine whether it is an example of a commutative ring or of a ﬁeld. Look at your data from the last two problems and this one. Any observations or conjectures? What might you expect to see for other Zm systems? 11. We can use the idea of “modulo m” to think about any integer and its remainder when divided by m. For example, a modulo 4 (also written as a (mod 4)) is the remainder you get after dividing a by 4. Notice that this result is an element of Z4 . Now take an integer a and square it. What are the possible values for a (mod 4)? What are the possible values for a (mod 8)? 12. A little research... (Internet sources like Wikipedia or Wolfram Mathworld or Planet Math are ﬁne, just please cite your sources.) (a) The size of a ﬁnite group is called its order. What is the other non-abelian group of order 8 which is not D8 ? How can you explain its structure? (If you have taken multivariable calculus, try to think about the cross product of unit vectors i, j and k in three-dimensional Euclidean space.) (b) I mentioned that all ﬁnite groups had essentially been classiﬁed by the 1980’s. One elusive example is the so-called “Monster Group.” Roughly how large is it? When was it discovered? What kind of group is it? (Extra questions to explore, research, ponder...) These are some of the other “open” questions we left class with... • Are there any non-abelian groups of prime order? • Are there any non-abelian groups of odd order? (Recall that “abelian” is the same as “com- mutative”). • Where do order of operations come from? Are they properties of Z? • Can we make a group out of the set of permutations of the numbers in {1, 2, 3}? Is this related to D6 ?