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Date
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Assignment |
| 3 Sep | p. 31 #1.4, 1.7, 1.10, 1.13, 2.6, 2.7, 2.8, 3.1, 3.2, 4.3, 4.4, 4.6, 5.1, 5.2, 5.3, 6.2, M.7, M.9 |
| 17 Sep |
p. 69 #1.2, 2.1, 2.3, 2.4, 2.5, 3.1, 4.1, 4.2, 4.3, 4.5, 4.6, 4.8, 5.1, 5.4, 5.5, 6.1, 6.4, 6.6, 6.7, 6.10, 7.1, 7.3, 7.4, 8.1, 8.2, 8.3, 8.6, 8.10, 10.2, 10.3, 10.4, 11.3, 11.6, 12.2, 12.4, M.2, M.6, M.13
Hints:
- For 8.3, try induction.
- For 10.2, you actually need not use anything in §10.
- For 11.6, use several different parts of Proposition 2.11.4.
- I may have other hints, if necessary.
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| 10 Oct |
p. 188 #1.1, 3.1, 3.2, 3.4, 3.6, 4.1, 4.2, 5.1, 5.2, 5.11, 6.1, 6.2, 6.3, 7.1, 7.7, 8.1, 8.2, 8.3, 9.1, 10.1, 10.2, M.2, M.3
Hints:
- For 3.6, use the properties given at the top of p. 160.
- For 5.2, think carefully about the consequences of not being a lattice (what kind of rotations can it have?), and whether it has any symmetry by reflection. Read p. 173 carefully if you need a little more help after doing that.
- For 5.3, write \(a\) and \(b\) in terms of \(a'\) and \(b'\), and vice versa. Use these expressions to build a matrix equation that gives you what you need.
- For 5.11(b), there is nothing special about the square root of 2, nor do you even need an irrational number. You can prove this for 1 and any rational number that is not also an integer.
- For 6.1, use the guidelines on p. 173.
- For 6.2, it helps if you first determine which symmetries generate the others.
- For 10.2, let \(x\) and \(y\) be two arbitrary elements of \(S\). Define \(X = \{ gU:g\in G \textrm{ and } x\in gU\}\), and \(Y=\{gU:g\in G \textrm{ and }y\in gU\}\). Find a one-to-one, onto map of elements of \(X\) to elements of \(Y\). It helps that \(G\) is transitive, because that means there is some \(g\in G\) such that \(gx=y\).
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| 24 Oct |
p. 221 #1.1, 1.2, 2.1, 2.2, 2.4, 2.5, 2.7, 2.9, 2.13, 2.17, 3.1, 3.2, 3.3, 4.1, 4.2, 5.2, 5.6, 5.7, 6.1, 6.4, 7.1, 7.2, 7.3, 7.4, 8.1, 8.3, 8.4, M.1, M.5, M.6
Hints:
- For 2.4, you can save a lot of time if you don't repeat work in orbits.
- For 2.13, the fact that \(N=5\) means it has a generator. I found it helpful to experiment with conjugating the generator.
- For 2.17, part of a solution is similar to the proof of Proposition 7.3.1. This actually solves the problem for a large class of groups; to finish, work with a quotient group, and exploit Lagrange's Theorem.
- For 3.2, you do not need any of the theory of \(p\)-groups.
- For 3.3(c), you should be able to write the class equations in terms of \(p\) alone -- no other variables are necessary.
- For 4.2, use Lemma 7.4.2. Show that if a normal subgroup \(N\) contains a transposition, then it contains all the transpositions, which means it is actually \(S_5\). Then show that if \(N\) contains any odd permutation at all, it also contains a transposition.
- For 5.7, show that any normal subgroup contains a 3-cycle. The proof of Theorem 7.5.4 comes in handy.
- For 6.4, keep in mind that the size of the conjugacy class must divide the order of the group, and every normal subgroup contains the conjugacy classes of its elements.
- For 7.1, be aware that the problem has a typographical error; the author means equation (7.7.1). Use the formula of Lemma 7.7.10, along with Wilson's Theorem, on p. 99. You need not prove Wilson's Theorem; if you want to see a proof, take Number Theory.
- For 7.4, a combination of the Class Equation, Equation 7.6.2, and Sylow Theorems will do the trick.
- For 8.1, it's not enough to identify the group isomorphic to \(S_3\times C_2\); you need to explain why. I found a complicated homomorphism; I'd award bonus points to anyone who finds a more elegant way to explain the isomorphism.
- For 8.4, look at the group generated by \(xy\). What can its order be?
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| 26 Nov |
p. 354 #1.2, 1.6, 1.7, 1.8, 1.9, 2.1, 2.2, 3.1, 3.2, 3.3, 3.4, 3.8, 3.9, 3.10, 3.12, 3.13, 4.1, 4.2, 4.3, 4.4, 5.1, 5.3, 5.4, 5.5, 5.6, 8.2, 8.3
Hints:
- For 5.5, the Correspondence Theorem might help, but I found a fairly direct approach.
- For 5.6, use (a) to show (b), and (b) to show (c).
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