Isomorphism Definition and 321 Threads

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...

Hi, I have to find a vector space V with a real sub space U and a bijective linear map. Here my Ideas and my questions: If the linear map is bijective, than dim V = dim U Because U is a real sub space the only way to valid this constraint is if the dimension is infinity. I wrote...

Homework Statement Let G=[a] and G'= be cyclic groups of the same order. Show, that among the isomorphisms \theta from G to G', there is exactly one with \theta(a)=c if and only if c is a generator of G. Homework Equations The Attempt at a Solution I have managed to show the...

Given H,K and general finite subgroups of G, ord(HK) = [(ord(H))(ord(K))] / ord(H intersection K) I know by the first isomorphism theorem that Isomorphic groups have the same order, but the left hand side of the equation is not a group is it? I am struggling to show this.

How the group of symmetries of the regular pentagram is isomorphic to the dihedral group of order 10? Suggest me some good basic isomorphism online tutorial.

Z_2[u]/<u^4+u+1> isomorphism Z_2[u]/<u^4+u^3+u^2+u+1> Homework Statement How to figure an isomorphism from Z_2[u]/<u^4 + u +1> to Z_2[u]/<u^4 + u^3 + u^2 + u + 1> What I can now show (after a page and a half of work) is that the two polynomials generating the ideals are irreducible...

Homework Statement can someone explain the 1st isomorphism theorem to me(in simple words) i really don't get it Homework Equations The Attempt at a Solution

The question is this: How many isomorphisms f are there from G to G' if G and G' are cyclic groups of order 8? My thoughts: Since f is an isomorphism, we know that it prserves the identity, so f:e-->e', e identity in G, e' identity in G'. Also f preserves the order of each element...

Homework Statement Let G be the group of real numbers under addition and let N be the subgroup of G consisting of all the integers. Prove that G/N is isomorphic to the group of all complex numbers of absolute value 1 under multiplication. Hint: consider the mapping f: R-->C given by...

What would be the technique to show A is isomorphic to (B intersection C)?where A, B and C are groups.

Homework Statement http://img219.imageshack.us/img219/2512/60637341vi6.png Homework Equations I think this is relevant: http://img505.imageshack.us/img505/336/51636887dc4.png The Attempt at a Solution A topological isomorphism implies that T and T-1 are bounded and given is that all cauchy...

I'm lost and don't even know where to start. Let F = Z mod 2, show F[x]/<x^3+X+1> and F[x]/<x^3+X^2+1> are isomorphic. I guess fist I need help understanding what those two factor rings look like and what elements they contain. Thanks

Homework Statement Let N = AB, where A and B are positive integers that are relatively prime. Prove that ZN is isomorphic to ZA x ZB. The attempt at a solution I'm considering the map f(n) = (n mod A, n mod B). I've been able to prove that it is homomorphic and injective. Is it safe to...

Homework Statement http://img297.imageshack.us/img297/1434/25931863lt2.png Homework Equations http://img297.imageshack.us/img297/4654/35953374xl9.png I don't see how you can show that T and T-1 are bounded. Furthermore I don't understand the notation Tf is that T*f or Tf as...

Homework Statement Is there an isomorphism from <R,+> to <R+,\times> where \phi(r)=0.5^{r} when r \in R? [b]2. Homework Equations For an isomorphism I know it is necessary to show there is a 1-1 and onto function. I am unsure if I can use the steps I am trying to use to show it is 1-1...

Given two left adjoints F,H:\mathcal{C}\to\mathcal{D} of a functor G:\mathcal{D}\to\mathcal{C}, how do we show that F and H are naturally isomorphic? This is my idea so far (I am working with the Hom-set defenition of adjunction): We need to construct a natural isomorphism \alpha. So, for...

I am trying to prove that the additive groups \mathbb{Z} and \mathbb{Q} are not isomorphic. I know it is not enough to show that there are maps such as, [tex]f:\mathbb{Q}\rightarrow \mathbb{Z}[/itex] where the input of the function, some f(x=\frac{a}{b}), will not be in the group of integers...

Homework Statement Prove that Q[x]/\langle x^2 - 2 \rangle is ring-isomorphic to Q[\sqrt{2}] = \{a + b\sqrt{2} \mid a,b \in Q\}. The attempt at a solution Denote \langle x^2 - 2 \rangle by I. a_0 + a_1x + \cdots + a_nx^n + I belongs to Q[x]/I. It has n + 1 coefficients which somehow map...

Hey Guys ; I need to discuss this problem with you. 1st of all , I'm going to post some posts about some questions with answers . ======================================================================= Q) Could we define a multiplication operation on \mathbb R^3 to have a field on it ...

I came across this problem today and haven't been able to figure it out... Give an example of a vector space V which isomorphic to a proper subspace W, i.e. V != W. It seems to me that V can't have a finite basis, but can't think of any examples regardless...any thoughts?

I'm mainly hoping that somebody else might have done the same exercise earlier. In that case it could be possible to spot where I'm going wrong. Homework Statement I'm supposed to prove that Lie algebras \mathfrak{o}(3) and \mathfrak{sp}(2) are isomorphic. Homework Equations Let's...

Hi everyone. I am new to these forums. I do Computer System Engineering at Brunel university in London. I did Maths and Physics at A-level but I'm struggling with some of the maths in my Engineering Maths module. Could someone please help me with the exam question I have attached with this post...

Can anyone tell me clearly what the criteria for isomorphism in linear algebra is? For instance, my book gives the following reason: Transformation T is not isomoprhic because T((t-1)(t-3)) = T(t^2 - 4t +3) = zero vector. I don't get why this means T is not an isomorphism. Can anyone...

Homework Statement Let V = P2(R) be the vector space of all polynomials P : R −> R that have order less than 2. We consider the mapping F : V −> V defined for all P belonging to V , by F(P(x)) = P'(x)+P(x) where P'(x) denotes the first derivative of the polynomial P. Question is: Show...

Show that the set S = { 0, 4, 8, 12, 16, 24} is a subring of Z subscript 28. Then prove that the map Ø: Z subscript 7 → S given by Ø(x) = 8x mod 28 is an isomorphism

[SOLVED] group theory Homework Statement My book says that Z_3 cross Z_4 is isomorphic to Z_12, which I am confused about because Z_3 cross Z_4 has four different generators and Z_12 only has 1. EDIT: wait that is not true, Z_12 has generators 1,5,7,11 It is probably true in general that the...

Does anybody know of a nice, intuitive way to remember the second and third isomorphism theorems?

Can someone explain isomorphism to me, with respect to vector spaces. Thanks!

Hello. My book offers this statement with no proof, i have been searching in other books with no luck ! I'm beginning to question whether or not the statement is valid at all ! Here it goes: "Every group G of order n is isomorphic to a subgroup of GLn(R)" Could someone please help me out...

I think I am missing a key info below. I have listed the problem statement, how I am approaching and why I think I am missing something. Please advise why I am wrong. Thanks Asif ============ Problem statement: Let T: U->V be an isomorphism. Let U1, U2,...,Un be linearly...

Just wondering if there is a general way of showing that (Z, .)n isomorphic to Zm X Zp with the obvious requirement that both groups have the same order?

Homework Statement Decide whether each map is an isomorphism (if it is an isomorphism then prove it and if it isn’t then state a condition that it fails to satisfy). Homework Equations f : M2×2 ---- P^3 given by: a b c d --- c + (d + c)x + (b + a)x^2 + ax^3 The Attempt at...

I am having a very hard time with a general concept of proving something. If I have some arbitrary function mapping one ring, let's say R, to another ring, S, and want to prove that R is isomorphic to S, then I need to show that there exists a bijective homomorphism between R and S. But how do I...

Hi. Hoping a could have a little bit of guidance with this question Show that U(8) is not isomorphic to U(10) So far, I've realized that in U(8) each element is it's own inverse while in U(10) 3 and 7 are inverses of each other. I guess that's really all I need to say that they aren't...

Abstract Algebra -- isomorphism question If N, M are normal subgroups of G, prove that NM/M is isomorphic to N/N intersect M. That's how the problem reads, although I am not sure how to make the proper upside-down cup intersection symbol appear on this forum. Or how to make the curly "="...

Suppose A\subset\mathfrak{g} and I\subset\mathfrak{g} are subalgebras of some Lie algebra, and I is an ideal. Is there something wrong with an isomorphism (A+I)/I \simeq A/I, a+i+I=a+I\mapsto a+I, for a\in A and i\in I? I cannot see what could be wrong, but all texts always give a theorem...

Hi, I am trying to prove a claim about order isomorphisms (similarity) between well ordered sets. I have an argument for it, but it seems needlessly complicated and I was wondering if anyone might have a simpler proof. Before stating the claim and my proof, I will define a few things: 1. A...

Hi everyone. I have two questions that I hope you can help me with. First when trying to show isomorphism between groups is it enough to show that the order of each element within the group is the same in the other group? For example the groups (Z/14Z)* and (Z/9Z)*. They are both of order...

Let k be a positive integer. define G_k = {x| 1<= x <= k with gcd(x,k)=1} prove that: a)G_k is a group under multiplication modulos k (i can do that). b)G_nm = G_n x G_m be defining an isomorphism.

Homework Statement Let G be a group with a normal subgroup N and subgroups K \triangleleft H \leq G. If H/K is nontrivial, prove that at least one of HN/KN and (H\cap N)/(K\cap N) must be nontrivial. Homework Equations The Three (or Four) Isomorphism Theorems. The Attempt at...

Homework Statement Prove that Aut(Z_2\oplus Z_4) \cong D_8 Homework Equations The Attempt at a Solution To start, I wrote out all of the elements of Z_2\oplus Z_4. There are 8 of them, of course. Then I need to find the automorphisms of it. It looks to me like they would be the same as...

Homework Statement Take L = \left(\begin{array}{ccc}0 & -a & -b \\b & c & 0 \\a & 0 & -c\end{array}\right) where a,b,c are complex numbers. Homework Equations I find that a basis for the above Lie Algebra is e_1 = \left(\begin{array}{ccc}0 & -1 & 0 \\0 & 0 & 0 \\1 & 0 &...

Show that the isomorphism O:L2(E)—>(E,E*) Where E is a vectorspace over a field K E* is a dual space L2:bilinear form L: n-linear form.

Homework Statement Let G_1 and G_2 be groups with normal subgroups H_1 and H_2, respectively. Further, we let \iota_1 : H_1 \rightarrow G_1 and \iota_2 : H_2 \rightarrow G_2 be the injection homomorphisms, and \nu_1 : G_1 \rightarrow G_1/H_1 and \nu_2 : G_2/H_2 be the quotient epimorphisms...

How do I find an isomorphism between Sn+m and Zn x Zm? provided n,m are not relatively prime? Thanks.

In general if two finite sets contain exactly the same number of unique elements than the two sets are isomorphic to each other. Is this correct? An isomorphism => both 1-1 and onto. If two sets both have an equal number of unique elements than they must be onto because every element in one set...

I am working through this algebra book and some of the problems. The chapter this comes out of is General Algebraic Systems and the section is Isomorphisms. I am new to proofs and maths higher than calculus I so I am not sure if I am following the text or not. There aren't any solutions and this...

Hi all, I wonder if there is an isomorphism between the group of \mathbb{N} and the group of \mathbb{Q} (or \mathbb{Q}+). I know there is a proof that there is a bijection between these sets, but I didn't find a way how to construct the isomorphism. What confuses me a little is that (I...

prove that there does not exist a homomorphism from G:= (integers modulo 8 direct product integers modulo 2) to H:= (intergers modulo 4 direct product integers modulo 4). Pf: i tried this route, assume that there is such a homomorphism. then by first isomorphism theorem, G/ker phi is...

Hi I'm trying to solve (find a group that is ismorphic to) (ZxZxZ)/<(2,4,8)>. (1,2,4)+<(2,4,8)> must be of order 2 in the factor group. (0,1,1)+<(2,4,8)> and (0,0,1)+<(2,4,8)> generates infinite cyclic subgroups of the factor group. So it would be reasonable to presume that (ZxZxZ)/<(2,4,8)>...