I haven't considered the possibility that the condition f(j+n)=f(j)+n forces bijectivity. But clearly the condition implies a bunch of things would not work: nothing of the form f(j)=mj where m>1, floor/ceiling functions, any functions which are constant between two 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
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...
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...
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...
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...
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...
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...
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...
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...
i've solved part b) and one direction of a); precisely the part that <x^3 + x> is contained in J but I have some difficulty proving the other inclusion. This is what I have so far: take any polynomial f in J, so f^2 is in I => so (x^4 + x^2) divides f^2 => x^2 divides f^2 (or even x divides f^2)...
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...
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...
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 +...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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'...
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...
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...