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    Function on Integers

    Homework Statement Is it possible to find a non-bijective function from the integers to the integers such that: f(j+n)=f(j)+n where n is a fixed integer greater than or equal to 1 and j arbitrary integer. Homework Equations The Attempt at a Solution
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    Constructing Path Connected Space

    Homework Statement 1. Construct a path connected space X such that the fundamental group of (X,x_0) where x_0 is the base point; such that the fundamental group is the symmetric group on 3 letters? 2. Let Z be the space obtained from a hollow cube by deleting the interior of its faces...
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    Computation of Ext group

    Homework Statement Compute Ext(R,Z) where R is the real numbers and Z is the integers Homework Equations The Attempt at a Solution This is part of a problem is given in Hilton-Stammbach page 109; the previous parts of the problem asks to compute Ext(Q,Z) and Hom(Q,Q/Z); I have...
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    Group of order 765 is abelian

    Homework Statement Show that the group of order 765 is abelian (Hint: let G act by conjugation on a normal Sylow p subgroup) Homework Equations Sylow theorems The Attempt at a Solution By using Sylow`s third theorem, I have calculated that the number of Sylow-3 subgroups and Sylow...
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    Group actions

    Homework Statement Let G be a group acting on a set X, and let g in G. Show that a subset Y of X is invariant under the action of the subgroup <g> of G iff gY=Y. When Y is finite, show that assuming gY is a subset of Y is enough. Homework Equations If Y is a subset of X, we write GY for...
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    Show G abelian

    Homework Statement Let G be a finite group and let I ={g in G: g^2 = e} \ {e} be its subset of involutions. Show that G is abelian if card(I) => (3/4)card(G). Homework Equations The Attempt at a Solution I don't really know how to proceed with this problem and to make use of 3/4. I know...
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    Showing the uniqueness of the group of integers

    Homework Statement Show that the infinite cyclic group Z is the unique group that is isomorphic to all its non-trivial proper subgroups Homework Equations The Attempt at a Solution Due to the fact that Z is cyclic and that every subgroup is a cyclic group, every subgroup of Z is a...
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    Complex analysis question

    Homework Statement if an entire function satisfies f(z+i)=f(z) and f(z+1)=f(z), must the function be constant? Homework Equations The Attempt at a Solution It's true that f(0) = f(k) = f(ik) where k is an integer. I'm wondering whether I can apply Liouville's theorem into this...
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    Show isomorphism between two groups

    Homework Statement Suppose G is a non-abelian group of order 12 in which there are exactly two elements of order 6 and exactly 7 elements of order 2. Show that G is isomorphic to the dihedral group D12. Homework Equations The Attempt at a Solution My attempt (and what is listed...
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    Principal Ideals

    Homework Statement Let I be an ideal of the commutative ring R, and let J = {y in R such that y^2 in I} a) If R is the polynomial ring Q[x] and I is the principal ideal of R generated by x^4 + x^2, show that J is the principal ideal of R generated by x^3 + x b) If R is a Principal Ideal...
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    Computing the conjugacy classes of D20(dihedral group of order 20)

    Homework Statement To compute the conjugacy classes of D20 Homework Equations The class equation, center of the group, group order, Lagrange's theorem that number of elements in conjugacy classes divides order of group The Attempt at a Solution i'm struggling to understand how to...
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    Conjugacy Class Problem

    Homework Statement If a group has order 20 has 4 conjugacy classes, it must have a trivial centre. True or False? Homework Equations The Class Equation The Attempt at a Solution I believed the answer to be false with this following counterexample: 20 = lZ(G)l + (20/4 + 20/4 + 20/5 +...
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    Proving irreducibility

    Homework Statement Let k be a field, and let f(x) = a_0 + a_1x +a_2x^2 +.......+a_nx^n in k[x] having degree n. If f(x) is irreducible, then so is a_n + a_n-1x+.....+a_0x^n Homework Equations The Attempt at a Solution A function that "reverses" the coefficients is not a...
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    Estimating partial derivatives/directional derivatives

    Homework Statement Let z = f(x,y) be a differentiable function on R^2 such that f(1, 2) = 3, f(1.2, 2.3) = 3.4 and f(0.9, 2.1) = 3.2. a) Estimate dz/dx and dz/dy at (1,2) (dz/dx and dz/dy are partial derivatives) b) Estimate the value of the directional derivative of z = f(x,y) at the...
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    Proving the identity of the field

    Homework Statement If S = {s in R such that s=/=1} is an abelian group under circle operation (Circle Operation a*b = a + b -ab for a, b in R) then R is a field Homework Equations The verification of the field axioms The Attempt at a Solution The field axiom that I'm struggling to...
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    Ring theory problem

    Homework Statement Let R be a ring that satisfies a^2 = a for all a in R. Prove that R is a commutative ring Homework Equations The Attempt at a Solution My attempt at this solution is (ab-ba)^2 = (ba-ab)^2 is true for any ring R => (ab-ba) = (ba - ab) => 2ab = 2ba => ab = ba. The...
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    Show R is not a domain

    Homework Statement Show that R = {0.5(a+b(root2)) : a,b are integers} is not a domain [b]2. Homework Equations A commutative ring (non trivial) is a domain iff (ab=0 => a=0 or b=0) The Attempt at a Solution I've try to construct a counterexample where x, y in R and xy=0 but x,y=/=0...
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    Non simplicity of a group

    Homework Statement Let G be a group with |G|= mp where p is prime and 1<m<p. Prove that G is not simple Homework Equations The Attempt at a Solution I have proven the existence of a subgroup H that has order p(via Cauchy's Theorem), but I don't know how to use the representation...
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    Real analysis problem

    Homework Statement Let E be nonempty subset of R which is bounded above (thus, a = sup E exists) Does there exist a strictly monotone sequence in E which converges to a? Homework Equations The Attempt at a Solution I've been thinking about just taking a monotone bounded (this...
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    Subsequential limit question

    Homework Statement Suppose {a_n} is a bounded sequence who's set of all subsequential limits points is {0,1}. Prove that there exists two subsequences, such that: one subsequence converges to 1 while the other converges to 0, and each a_n belongs to exactly one of these subsequences...
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    Needs a counterexample for homomorphisms

    Homework Statement Let A, B be groups and A' and B' be normal subgroups of A and B respectively. Let f: A --> B be a homomorphism with f(A') being a subgroup of B'. There is a well-defined homomorphism g: A/A' -----> B/B' defined by g: aA' ---> f(a)B' Find an example in which f is...
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    Intersection of Groups

    Homework Statement If H, K are subgroups of G, show that H intersect K is a subgroup of H Homework Equations I know that H intersect K is a subgroup of G; I proved this already but I'm wondering how H intersect K is a subgroup of H The Attempt at a Solution I'm quite sure this is true...
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    Modular arithmetic (casting out 9's)

    Homework Statement Let m be a positive integer and m' be an integer obtained from m by rearranging its digits. Prove that m-m' is a multiple of 9 Homework Equations Casting out 9's method The Attempt at a Solution So I found that by applying the casting out 9's method on m and m'...
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    Introductory abstract algebra question (computing permutations)

    Homework Statement Let A, B be permutations and A = (1 3 5 10)(3 15 8)(4 14 11 7 12 9) and B = (1 14)(2 9 15 13 4)(3 10)(5 12 7)(8 11) Find AB. Homework Equations The Attempt at a Solution I am struggling with finding the product of this permutations and can't quite get the...
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    Very Simple Physics Problem

    i have just started physics in high school and i got this problem i want some help on this question: A small fish is dropped by a pelican that is rising steadily at 0.41m/s. a)After 1.8s, what is the velocity of the fish? Answer in units of m/s b)How far below the pelican is the fish after...
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