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...