Isomorphism Definition and 321 Threads

I need to find an isomorphism between the group of orientation preserving rigid motions of the plane (translations, rotations) and complex valued matrices of the form a b 0 1 where |a|=1. I defined an isomorphism where the rotation part goes to e^it with angle t and the translation...

I am trying to prove that T'' = T (where T'' is the double transpose of T) by showing that the the dual of the dual of a linear finite vector space is isomorphic to the original vector space. i.e., T: X --> U (A linear mapping) The transpose of T is defined as the following: T'...

GroupTheory - Isomorphisms Hey I'm stuck on these 2 questions, was wondering if anyone could assist me: Let G be a nontrivial group. 1) Show that if any nontrivial subgroup of G coincides with G then G is isomorphic to C_p, where p is prime. (C_p is the cyclic group of order p!) 2) Show...

Sets in Linear Space I am trying to show the set of all row vectors in some set K with dimension n is the same as the set of all functions with values in K, defined on an arbitrary set S with dimension n. I am using isomorphism to show this, but I can't determine how to show that the...

I'm slightly confused with the following function so I was wondering if anybody could give me some hints as to the next step. A function f is defined as f:\mathbb{C} \longrightarrow \mathbb{C} \\ ~~z \longmapsto |z| where \mathbb{C} = (\mathbb{C},+) assuming the function is...

A Group Problem :) So.. I have to demonstrate that those following two groups are isomorfic , that there is an isomorphism between those 2 groups : Now I know that in such a way that an isomorphism might be , there must also be defined a function in G with values in Z3, and is soooo...

I wondered if someone could help me with the following problem. Gn (n >= 2) is a graph representing the vertices abd edges of a regular 2n sided polygon, with additional edges formed by the diagonals for each vertex joined to the vertex opposite i.e. vertex 1 is joined to n+1, vertex 2 to n+2...

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...