Isomorphism Definition and 314 Threads

Homework Statement Let H and K be normal Subgroups of a group G s.t H intersect K = {e}. Show that G is isomorphic to a subgroup of G/H + G/K. Homework Equations G/H+G/K= direct product of G/H and G/K. The Attempt at a Solution Proof/ Lets define are mapping f:G to G/H+G/K by...

Hi, everyone: A couple of things, please: 1) I am going over the Leray-Hirsch theorem in Hatcher's AT , which gives the conditions under which we can obtain the cohomology of the top space of the bundle from the tensor product of the cohomology of the fiber, and that of the base (...

Homework Statement I have to show that \sum ai xi -> (a0 \sum ai) is a ring homomorphism from C[x] to C x C I then have to use the first isomorphism theorem to show that there is an isomorphism from C[x]/ (x(x-1)) to C x C where (x(x-1)) is the principal ideal (p) generated by the element...

Homework Statement I have to use the first isomorphism theorem to determine whether C16 (cyclic group order 16) has a quotient group isomorphic to C4. Homework Equations First isomorphism theorem The Attempt at a Solution C16 = {e, a, ..., a^15} C4 = {e, b, ..., b^3}...

So it says here "Let S be a set of sets. Show that isomorphism is an equivalence relation on S." So in order to approach this proof, can I just use the Reflexive, Symmetrical, and Transitive properties that is basically the definition of equivalence relations? eg. suppose x, y, z are sets...

I can't seem to find any sort of concrete definition anywhere... it always seems a bit hand wavy :( In particular I want to know what is an isomorphism between two banach algebras? Thanks.

Hi all, If I have to prove that the graph G and its complement G' are isomorphic, then is it enough to prove that both G and G' will have the same number of edges. Intuitively its clear to me, but how do I prove this. If there's a counterexample, please post. Thanks in advance.

Homework Statement If G is a group and a[SIZE="4"]ϵG, then the inner automorphism θa: G --> G is defined by θa(g) = aga-1. Let Inn(G) = group of inner automorphisms and Z(G) = the centre of G. Use the Isomorphism theorem to show G/Z(G)[SIZE="3"]≅Inn(G). Homework Equations The...

Homework Statement Let V be a vector space over the field of complex numbers, and suppose there is an isomorphism T of V onto C3. Let a1, a2, a3,a4 be vectors in V such that Ta1 = (1, 0 ,i) Ta2 = (-2, 1+i, 0) Ta3 = (-1, 1, 1) Ta4 = (2^1/2, i, 3) Let W1 be the suubspace spanned by a1...

I've recently encountered some forms of the second and third isomorphism theorem, but I don't quite get them. Could anyone explain in a bit of details please? I guess my thought was not in the right direction or something. (Second isomorphism theorem) Let A be a subring and I an ideal of the...

I think I've solved this problem, but the examples in my textbook are not giving me any indication as to whether my reasoning is sound. Homework Statement Is the transformation T(M) = M\left[ \begin{array}{cccc} 1 & 2 \\ 3 & 6\end{array} \right] from \mathbb{R}2x2 to \mathbb{R}2x2 linear...

Homework Statement Prove taht if the order n of a group G is a prime number, then G must be isomorphic to the cyclic group fo order n, C_n. The Attempt at a Solution We have previously proven that a group can can be written as S = \{A,A^2,A^3,A^4...,A^n = E\} where E is the identity and the...

Find an isomorphism from the subgroup of GL2(C) of the form \begin{pmatrix} a & b\\ 0 & 1 \end{pmatrix} ,\left | a \right |=1 to the group of orientation preserving rigid motions. *The problem is from Artin's Algebra Chapter5

Homework Statement let G be a group and let g be one fixed element of G. Show that the map ig, such that ig(x) = gxg' for x in G, is an isomorphism of G with itself.Homework Equations The Attempt at a Solution not even really understanding the question. can someone break it down for me, and...

Homework Statement Suppose f and g are isomorphisms from U to V. Prove of disprove each of the following statements: a) The mapping f + g is an isomorphism from U to V. Homework Equations The Attempt at a Solution I have no idea where to start.. do I need to show that f and...

Homework Statement Determine whether the following mappings f is onto or one-to-one. Is f an isomorphism? a) f maps R2 into R2 and is defined by f(x,y) = (x-2y, x+y) b) f maps R2 into R3 and is defined by f(x,y) = (x, y, x+y) i) f maps R3 into P2(R), defined by f(a1, a2, a3) = a2 - a3x +...

This statement was made in my class and I'm trying still to piece together the details of it... We say that some rational polynomial, f has a Galois group isomorphic to the quaternions. We can then conclude that the polynomial has degree n \geq 8. I have a few thoughts on this and I might...

Homework Statement Prove that there is no isomorphism, \phi, from Q under addition to R under addition Homework Equations An isomorphism \phi:Q to R is a bijection such that \phi(x + y) = \phi(x) + \phi(y), where x,y are elements of Q \phi(0) = 0. \phi(-x) = -\phi(x) The...

How many graphs(non isomorphic) can you construct from the degree sequence (3,3,3,3,4). The answer has to be proven of course. The only one I could find was a W5 graph, but i can't prove that it is the only one. I know that for two graphs to be isomorphic, a bijection has to exist between the...

Homework Statement If G contains a normal subgroup H which is isomorphic to \mathbb{Z}_2, and if the corresponding quotient group is infinite cyclic, prove that G is isomorphic to \mathbb{Z}\times\mathbb{Z}_2 The Attempt at a Solution G/H is infinite cyclic, this means that any g\{h1,h2\} is...

Define T: F^2 --> P_1(F) by T(a, b) = a + bx (with P_1 denoting P sub 1) I usually prove problems such as this by constructing a matrix of T using bases for the vector spaces and then proving that the matrix is invertible, but is the following also a viable proof that T is an isomorphism...

Given: G is the group of matrices of the form: 1 n 0 1 Where n is an element of Z, and G is a group under matrix multiplication. I must show that G is isomorphic to the group of integers Z. I do not know how to do this, since all examples we covered gave us the specific mapping...

HELP! Find all abelian groups (up to isomorphism)! I am really confused on this topic. can you give me an example and explain how you found, pleaseee! for example, when i find abelian group of order 20; |G|=20 i will find all factors and write all of them, Z_20 (Z_10) * (Z_2) (Z_5)*...

Homework Statement Show that II18 / <3> is isomorphic to II3. Homework Equations II18 = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18} <3> = {0,3,6,9,12,15} II3 = {1,2,3} II18 / <3> = {3,6,9,12,15,18} The Attempt at a Solution

Let G be a finite group. For all elements of G (the following holds: g^2=e(the idendity.) So , all except the idendity have order two. Proof that G is isomorphic to a finite number of copies of Z_2 ( the group of adittion mod 2, Z_2 has only two elements (zero and one).) I can try to tell...

Let G be a group and let \phi be an isomorphism from G to G. Let H be a subgroup. Hint: These subgroups should already be familiar to you. Let H={z in C:\phi(z)=z} This would be the subgroup of {-1,1}, this would be the group {-1,1} under multiplication. Let H={z in C: \phi(z)=-z}...

I used to think 1 by 1 matrix is a scalar, but someone argued with me and said they were different. Then I tried to convince him that we actually couldn't find the difference between their fields. He then told me the fields were just isomorphic, so he still didn't agree with my opinion. I can't...

Homework Statement Show that Z/mZ X Z/nZ isomorphic to Z/mnZ iff m and n are relatively prime. (Using first isomorphism theorem) Homework Equations The Attempt at a Solution Okay, first I want to construct a hom f : Z/mZ X Z/nZ ---> Z/mnZ I have f(1,0).m = 0(mod mn) =...

Hello, I was doing self studying abstract algebra from the online lecture notes posted by Robert Ash and I hit against the following theorem. I am posting it in the topology section because without a geometric/topological meaning to the concept I am never able to understand the topic and that...

Homework Statement The problem is as follows: Let f : R^2 map to R^2 be rotation through an angle of theta radians about the origin. Prove that f is an isomorphism. Homework Equations Let f : R^2 \rightarrow R^2 The Attempt at a Solution I know that the rotation...

Directions: Let \phi: G \rightarrow G' be an isomorphism of a group <G, *> with a group <G', *'>. Write out a proof to convince a skeptic of the intuitive clear statement. Problem: 41.) If H is a subgroup of G, then \phi[H] = {\phi(h)| h \in H} is a subgroup of G'. That is, an...

Hello, I just cracked open this abstract algebra book, and saw a problem I have no idea how to solve. Instruction: Determine whether the given map \phi is an isomorphism of the first binary structure with the second. If it is not an isomorphism, why not? (Note: F is the set of all functions...

Hey all, Okay, let me give this a wack. I want to show that A \times 1 is isomorphic to A. I'm aware that this is trivial, even for a category theory style. However, sticking to the defs and conventions is tricky if you aren't aware of the subtleties, which is why I'm posting this. So here...

Notations: V denotes a vector space A, B, C, D denote subspaces of V respectively ≈ denotes the isomorphic relationship of the left and right operand dim(?) denotes the dimension of "?" Question: Find a vector space V and decompositions: V = A ⊕ B = C ⊕ D with A≈C but B and D are not...

Homework Statement a belongs to R show that the map L: R^n------R^n>0 (R^n>0 denote the n-fold cartesian product of R>0 with itself) (a1) (...) ---------- (an) (e^a1) (...) (e^an) is a isomorphism between the vector space R^n and the vector space R^n>0 Homework Equations...

Homework Statement \phi:G-->G' Let \phi be an isomorphism. Prove that \phi maps the e identity of G to e', the identity of G' and for every a\inG, \phi(a^{-1})=^\phi(a){-1}. Homework Equations The Attempt at a Solution We have an isomorphism, therefore one to one, onto and has...

Homework Statement Let T: V \rightarrow Z be a linear transformation of a vector space V onto a vector space Z. Define the mapping \bar{T}: V/N(T) \rightarrow Z by \bar{T}(v + N(T)) = T(v) for any coset v+N(T) in V/N(T). a) Prove that \bar{T} is well-defined; that is, prove that if...

Can anyone help me with isomorphisms? I am to construct an isomorphism with 25 elements and I am very confused. Thanks!

Homework Statement Show that G = {[1 0 [-1 0 [0 -1 [0 1 0 1], 0 -1], 1 0], -1 0]} is a subgroup of GL[SUB]2[/SUB(Z) isomorphic to {1,-1,i,-i}. The Attempt at a Solution I am clearly sure each element in G can be denoted as {1,-1,i,-i}. (I can explain...

Homework Statement Show that the group Z/<(a,b)> is isomorphic to Z if gcd(a,b)=1. Find generators of Z/<(a,b)>. 2. Relevant information Please note that the question is asking for Z/<(a,b)>, not ZxZ/<(a,b)>. I am having trouble understanding the meaning behind <(a,b)> as a subgroup of...

Homework Statement Let T be defined on F^2 by (x1,x2)T=(w*x1+y*x2, z*x1+v*x2) where w,y,z,v are some fixed elements in F. (a) Prove that T is a homomorphism of F^2 into itself. (b) Find necessary and sufficient conditions on w,y,z,v so that T is an isomorphism. The Attempt at a Solution I...

Homework Statement There is an isomorphism of U_{7} with Z_{7} in which \zeta=e^{(i2\pi}/7\leftrightarrow4. Find the element in Z_{7} to which \zeta^{m} must correspond for m=0,2,3,4,5, and 6. Homework Equations The Attempt at a Solution \zeta^{0}=0 \zeta^{2}=4+_{7}4=1...

Determine whether the given map \varphi is an isomorphism of the first binary structure with the second. < M2(R ), usual multiplication > with <R, usual multiplication> where \varphi(A) is the determinant of matrix A. The determinant of the matrix is ad-bc, so \varphi(A)=ad-bc. For this...

I'm fairly certain the following is a vector space isomorphism \phi :\mathbb{R}^\infty\rightarrow\mathbb{R}^\infty where the vector space is the space of infinite sequences of real numbers and phi is defined by \phi(a_1,a_2,...)=(0,a_1,a_2,...) . The mapping is linear and the inverse seems to...

order isomorphism f:R-->R let f is order isomorphism from (R,<) to (R,<). why f is continuous ? so f is bijection and a<b <--> f(a)<f(b), so what ?

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.