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  1. O

    Solving a Differential Equation

    Homework Statement \[P^{'}(t)+(\lambda +\mu )P(t)=\lambda \] I have never worked with differential equations before and I am trying to work off of the one example we did in class, but I can't figure out where I am going wrong. Homework Equations The Attempt at a Solution...
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    Covariance of Binomial Random Variables

    Homework Statement Let X be the number of 1's and Y be the number of 2's that occur in n rolls of a fair die. Find Cov(X, Y) Homework Equations Cov(X,Y) = E(XY) - E(X)E(Y) The Attempt at a Solution Both X and Y are binomial with parameters n and 1/6. Thus it is easy to find E(X)...
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    Interpretation of random variable

    Homework Statement The probability mass function of a random variable X is: P(X=k) = (r+k-1 C r-1)pr(1-p)k Give an interpretation of X. Homework Equations The Attempt at a Solution The PMF looks like the setup for a binomial random variable. The first combination looks like you...
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    Probability of the Union of Indepedent Events

    Homework Statement Show that if A1, A2, ..., An are independent events then P(A1 U A2 U ... An) = 1 - [1-P(A1)][1-P(A2)]...[1-P(An)] Homework Equations If A and B are independent then the probability of their intersection is P(A)P(B). The same can also be said of AC and B. The...
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    Proving Non-Simplicity

    Homework Statement Let G be a finite group. Prove that if some conjugacy class has exactly two elements then G can't be simple Homework Equations The Attempt at a Solution I originally proved this accidentally assuming that G is abelian, which it isn't. So say x and y are...
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    Sylow Subgroups of Symmetric Groups

    Here is my idea although it may be way off the mark. Still looking at p=3, the subgroup could contain powers of 10 different 9-cycles that way you'd have 80 different elements plus the identity. I think that works. Although I guess we can't be sure that it's closed. Well, I'm basically out of...
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    Sylow Subgroups of Symmetric Groups

    Okay, that's true. Sorry. I guess I really just have no idea how to find generators for this. I was trying with p=3 for my example and found that 3^4 is the highest power of 3 that divides 9! since there are 4 factors of 3 in 9*8*7*6*5*4*3*2*1. So the Sylow p-subgroup would have 81 elements. So...
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    Sylow Subgroups of Symmetric Groups

    Sorry, it's supposed to be restricted to just odd primes, in which case I think that still holds, correct?
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    Sylow Subgroups of Symmetric Groups

    Homework Statement Find a set of generators for a p-Sylow subgroup K of Sp2 . Find the order of K and determine whether it is normal in Sp2 and if it is abelian. Homework Equations The Attempt at a Solution So far I have that the order of Sp2 is p2!. So p2 is the highest power of...
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    Finding Torsion Coefficients

    Homework Statement Let G be a finite abelian group and let #(n) signify the number of elements x in G which satisfy x^n = e. Find the torsion coefficients of G when #(2)=16, #(4)=32, #(3)=9, #(9)=81 and x^36=e for all x in G. Homework Equations The Attempt at a Solution I really...
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    Sylow Subgroups

    Homework Statement If J is a subgroup of G whose order is a power of a pirme p, prove that J must be contained in a Sylow p-subgroup of G. (Take H to be a Sylow p-subgroup of G and let X be the set of left cosets of H. Define an action of G on X by g(xH) = gxH and consider the induced action...
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    Finding a Rotational Symmetry Group

    Well, my guess is that the number of rotational symmetries for the new object is less than the number for the original dodecahedron. I would guess it's dihedral or icosahedral (since I know of one axis with order 5) symmetry even though I do see a couple of problems with those theories, but I...
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    Group Actions on Truncated Octahedron

    Maybe this doesn't make sense, but I don't see how the edges of the square can be in the same orbit as the edges that only border hexagons.
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    Group Actions on Truncated Octahedron

    Homework Statement Let G be the group of rotational symmetries of the octahedron and consider the action of G on the edges of the truncated octahedron. Describe the orbits of this action. Choose one representative element in each orbit. Describe the stabilizers of these representative...
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    Symmetry Groups and Group Actions

    Would the last two 3-cycles really be fixed? If there were six elements remaining to put into cycles there are still 20 different ways to put them into two separate 3-cycles, correct?
  16. O

    Symmetry Groups and Group Actions

    Well, I tried looking at it that way, but it seems really complicated. But here is what I have come up with, please let me know if I am missing something since I haven't done this type of thing in a while. 17 choose 4 = 2380 13 choose 4 = 715 9 choose 3 = 84 6 choose 3 = 20 3 choose 3 =...
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    Symmetry Groups and Group Actions

    Homework Statement I would like to find the number of distinct elements in S17 that are made up of two 4-cycles and three 3-cycles. Homework Equations The Attempt at a Solution This seems like a very simple question but since the group is so huge it's hard to figure out. I have...
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    Conjugacy Groups of A5

    Homework Statement I am interested in proving that A5 has no normal subgroups except itself and {e}. The Attempt at a Solution Some proofs that I have seen use centralizers to do this, but since I haven't gone through that yet I think there should be some say to do it without them. My...
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    Subgroups and LaGrange's Theorem

    Oh, okay. I see it now. Thanks very much.
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    Subgroups and LaGrange's Theorem

    Well, I suppose it would be a group if it contained the identity, g and the inverse of g. This is what I was trying to use to find a contradiction but I'm not sure how this leads to the fact that H and K are relatively prime.
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    Subgroups and LaGrange's Theorem

    Let H and K be finite subgroups of a group G whose orders are relatively prime. Show H and K have only the identity element in common. By LaGrange's theorem I know that the orders of H and K must divide the order of G. I have attempted a proof by contradiction but have had no luck arriving at...
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    Finding Isomorphisms

    Yes, thanks very much.
  23. O

    Finding Isomorphisms

    Okay. I see that, but I guess I'm not sure how to use that fact to show that G x Z is isomorphic to G. Should I try to show that G x Z is also isomorphic to the infinite direct product?
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    Finding Isomorphisms

    I have the group G whose elements are infinite sequences of integers (a1, a2, ...). These sequences combine termwise as such: (a1, a2,...)(b1, b2,...) = (a1+b2, a2+b2,...) I would like to find an isomorphism from G x Z (the direct product of G and the integers) to G as well as an isomorphism...
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    Showing Isomorphisms in Subgroups

    Okay, I see it now. Thanks.
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    Showing Isomorphisms in Subgroups

    But don't the generators of G have to lie in G? Because 1+0 is odd.
  27. O

    Showing Isomorphisms in Subgroups

    I see the visual representation, but I guess I am just not seeing how this gives you the generators. The only way I can think to generate G is with (2,0), (0,2) and (1,1). I'm also not sure how to map the rest of the elements that are not generators.
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    Showing Subgroups of a Permutation Group are Isomorphic

    Oops. I meant G=<(123), (123)(456)> and H=<(14), (123)(456)>.
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    Showing Isomorphisms in Subgroups

    Homework Statement Let G be a subset of Z x Z (direct product) where G = {(a,b)|a+b = 2k for some integer k}. I'd like to show that G is a proper subgroup of Z x Z and determine whether G is isomorphic to Z x Z. I am pretty sure I have shown that it is a proper subgroup but the isomorphism...
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    Showing Subgroups of a Permutation Group are Isomorphic

    Define two subgroups of S6: G=[e, (123), (123)(456)] H=[e, (14), (123)(456)] Determine whether G and H are isomorphic. It seems as if they should be since they have the same cardinality and you can certainly map the elements to one another, but I don't know what other factors need to be...
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