| HW1 #1 | You don't need to solve the other problems (#1-12,14), just use those results. It is a good idea in every chapter to read over all the problems, even those you aren't going to do, and try to understand them.
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| HW1 #6(a),(b) | The little numbers are exponents, and they mean "multiply this thing by intself a bunch of times". So, for example, (1 2 3 4)^2 = (1 2 3 4)(1 2 3 4) = (1 3)(2 4) | |||||||||||||||||||||||||||||||||||||||||||||
| HW1 #7 | A permutation of a set A is another word for a bijection from A to A. S a permutation in S_n is a bijection from {1,2,3,4,...,n} to itself.
I'll call tau T, and sigma S. The important point is to realize that T(a_1) is a *number*. So ( T(a_1) T(a_2) .... T(a_k) ) is a cycle, C. I.e. it's a function. So is TST^{-1}. And if you want to show that two functions are equal, you show that they agree on every number. I personally have no idea where C sends the number 1, but there is atleast one *number* whose image under C I know!
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| HW1 #9(c) | Euclid's sneaky trick was to say "Assume ..... and then let's write P=(p_1)(p_2..(p_k) + 1 ....... oh no! that's a new prime! ......" (The ... are the rest of his argument.) You should try to have the same structure to your argument, except replace every occurence of the word "prime" with "prime of the form 4n+3". You have to be even sneakier about how to invent the P (the way Euclid did was not sneaky enough.)
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| HW1 #2 | Remember that the things you are trying to show are equal (in (a) ) are SETS. The best way to show that two sets are equal is to show that they are contained in each other. You will do this in part (a) by unravelling the definitions... For example, to show f(A u B) is a subset of f(A) u f(B), you argue as follows: Assume y is in f(A u B). Then there is an x in A u B with y=f(x) (def'n of f(AuB)). Since x is in AuB, it's in A or it's in B (def'n of u). If x is in A, then y=f(x) is in f(A) (def'n of f(A)). Similarly, if x is in B, y is in f(A) u f(B). Therefore every element of f(AuB) is in f(A) u f(B), so we have shown that f(AuB) is a subset of f(A) u f(B). The next step is to show that f(A) u f(B) is a subset of f(AuB)...
HW2 #2
| Pointwise addition of functions means the following operation: For two functions f, g:R->R, (f+g) is the function with the property (f+g)(x) = f(x) + g(x) for all x. | HW2 #7
| A and B should be subgroups of S_3
| HW2 #8(b)
| The converse of "A implies B" is "B implies A"
| HW3 #1
| You'll need the division algorithm.
| HW3 #3
| The things that's so great about having an isomorphism between two groups is that you know that the two groups are "as close to being exactly the same as you could possible expect them to be". Put another way, the isomorphism is just a relabelling function that changes the names of the elements in the one group to the names of the elements of the other group, and changes the name of the operation, but doesn't change any of the actual structure of the group. So, for example, if in the domain group, you know that two elements commute: ab=ba, then after you apply an isomorphism, f, you must have f(a)f(b) = f(b)f(a). Notice that we've changed the name of 'a' to 'f(a)', and similarly for 'b', but we haven't changed the structural property of this group that the two elements commute. Similarly, if a happened to have order k, then (since that is a structural property of the element a, anf since f is an isomorphsim) f(a) would have order k too. SO: the best way to show that two groups ARE NOT ISOMORPHIC is to demonstrate that there is some structural property that they don't share: e.g. one is abelian and the other is not, one has a different number of elements from the other, one has exactly three elements of order 5 and the other has more etc.etc. A good problem for me to give on the exam might be something like "Show that if fG->G' is an isomorphism, and g in G has order k, then f(g) in G' has order k too. In other words, show that the order of an element is preserved under an isomorphism, i.e. that the order of an element is a tructural property of that element. | HW3 #4
| You'll need to know the definitions of "kernel", "homomorphism", "injection", and remember that a good way to argue an "if and only if" statement is to attack each direction of the implication separately. | HW3 #6
| Since all subgroups are normal, you just need to figure out which is the best subgroup for you to think about! (What is Jameel's favorite kind of subgroup? | HW3 #7
| (a) Use only the definition of normal and the definition given in the problem. (b) The elements of the group G/Z(G) are the cosets of Z(G) in G. What are some good names for these cosets? (In other words, they look like xZ(G), yZ(G),..., but each of these cosets has lots of possible names, and from the info in the problem we can pick some better names.)
| HW3 #8(b)
| If you have a group G with a subgroup H, and you want to show that the quotient G/H is isomorphic to some other group G', a very good strategy is to find an ONTO homomorphism fro G to G' that has kernel H. Then, but what we did today in class, you get an isomorphism G/H ---> G' for free!
| HW3 #9(d) | (This problem is optional.) Think about trying to use the 1st homomorphism theorem on a morphism from H to HN/N. | HW6#1
| In a few of these parts, you will know automatically that the structure defined is a subgroup - because of some theorem or exercise we did weeks ago in the group theory unit. If you don't see why this is, it's a good exercise anyway to go through the motions to remember the axios for a group.
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HW6#4
| A good strategy might be to think about quotient rings.
HW6#3(b)
| Once you have shown that the homomorphism is 1-1, you get onto for free: any one-to-one map between finite sets with the same number of elements is automatically onto.
| HW6#2
| A good strategy would be to find a couple of generators for the domain, and try to define your homomorphism by assigning a value in the target space to each generator... What restrictions must you put on the images of the generators you chose? HW6#5(d)
| This should be a lot like finding the inverse of [k]_p in Z/pZ. By ``[x^2-2x + 1]'' I mean the coset this polynomial represents, or its equivalence class mod the ideal in the quotient.
| HW8#5(a)
| What kind of structure is F_p[x]/(f)? What does this tell you about the roots of f compared to the polynomial x^q-x?
| HW8#1
| Check out the last problem on HW7 for a good starting point. HW8#3
| Check out the second paragraph on page 223 of Herstein.
| HW8#5(b)
| It might help that (y-1)[y^(r-1)+y^(r-2)+...+y+1] = y^r -1. |