Isomorphism Definition and 314 Threads

Problem: " Prove (Third Isomorphism THeorem) If M and N are normal subgroups of G and N < or = to M, that (G/N)/(M/N) is isomorphic to G/M." Work done so far: Using simply definitions I have simplified (G/N)/(M/N) to (GM/N). Now using the first Isomorphism theorem I want to show that a...

Prove that there exists a group isomorphism between (Q&,*) and (Z[X],+) where Q& is the set of strictly positive rational numbers. I was thinking of mapping a p_n, being the nth prime in Q& to x^(n-1). Would this work for this case?

what is the easiest way to show that Q[x]/<x^2-2> is ring isomorphic to Q[sqrt2]={a+b(sqrt2)|a,b in Q} just give me a hint how to start

"Let R be the ring Zp[x] of polynomials with coefficients in the finite field Zp, and let f:R->S be a surjective homomorphism from R to a ring S. Show that S is either isomorphic to R, or is a finite ring." If S is isomorphic to R, then we're done. If S is not isomorphic to R, then by...

Can isomorphism of matrix groups \phi: G_1 \rightarrow G_2 always be expressed by \phi(M) = S M S^{-1}?

Am I doing this right? I'd appreciate any feedback. Let T:U ---> v be an isomorphism. Show that T^-1: V----> U is linear. i. T^-1(0) = 0 ii. T^-1(-V) = -T^-1(V) T^-1(-0) = T^-1(0+0) = T^-1(0) + T^-1(0) T^-1(0) = 0 T^-1(-V) = T^-1((-1)V) =(-1)T^-1(V) = -T^-1(V)...

ok, I've pasting some of the stuff I've done in scientific workplace 3.0. should be easier to read than in plain text. hope some of you can help me... just ask if there is something you don't get. I am supposed to prove that $Map(n,K)\thickapprox K[X_{1},..X_{n}]/I.$ where I is the ideal...

I'm looking for help constructing the natural isomorphism between V\otimes V^* and \operatorname{End}(V), with V a vector space. So far, I think I should have functors F and G which take V \mapsto V\otimes V^* and V \mapsto \operatorname{End}(V). I'm having a little trouble figuring out how...

how do i prove that Aut n(K) is isomorphic to the symmetric group Sq^n. K is a finite field of q elements. Aut n(K) is the group of polynomial automorphisms over K. n i just the number of variables/indeterminants. so i guess i have to somehow show that Aut n(K) are the permutations of...

Let V denote the vector space that consists of all sequences {a_n} in F (field) that have only a finite number of nonzero terms a_n. Let W = P(F) (all polynomials with coefficients from field F). Define, T: V --> W by T(s) = sum(s(i)*x^i, 0, n) where n is the largest integer s.t. s(n) !=...

How would I go about proving the following: If G has an element of order n, then H has an element of order n. I am not sure how to start, if I should some how go about proving one to one and onto. Help

let G be an abelian group, and n positive integer phi is a map frm G to G sending x->x^n phi is a homomorphism show that a.)ker phi={g from G, |g| divides n} b.) phi is an isomorphism if n is relatively primes to |G| i have no clue how to even start the prob...:-(

i can't grasp these concepts, 1-to-1 and onto have always annoyed me. here's 1 question, (i don't know how to post symbols so Beta ..) (C is Complex numbers) Let Beta:<C,+> -> <C,+> by Beta(a+bi)=a-bi (that is, the image is a +(-b)i). Prove Beta is an isomorphism of...

find an isomorphism from from the group of integers under addition to the group of even integers under addition. I know, very simple question, but I don't know what I am doing here... the hint in the book says to try n to 2n. I thought of that too, since it specificaly says integers to...