Recent content by playa007

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 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 SolutionI don't really know how to proceed with this problem and to make use of 3/4. I know that...

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 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 well-defined...

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...