Isomorphism Definition and 314 Threads

Homework Statement Homework Equations The Attempt at a Solution

what does it really mean? for instance if asked to list all abelian grps of order 12 up to iso, then do we include Z12 or not?

All groups are finite abelian if K⊕K ≅ N⊕N, prove that K≅N I'm thinking of constructing bijection, but I don't know if my argument makes sense! since K⊕K ≅ N⊕N, there exists a bij between the two assume ψ: K⊕K ----> N⊕N (k,k') |---> (n,n') where n = f(k) for some...

Homework Statement We're looking at a mapping from P2 (polynomials of degree two or less) to M2(R) (the set of 2x2 real matrices). The nuance here is that the transformation into the matricies is such that its basis consists of only three independent matrices, making its dimension 3. This...

Homework Statement I am given the group (G,\bullet) consisting of all elements that are invertible in the ring Z/20Z. I am to find the direct product of cyclic groups, which this group G is supposedly isomorphic to. I am also to describe the isomorphism. Homework Equations The...

Homework Statement Let U be a finite dimensional vector space and suppose that U and W are nonzero subspaces of V prove that (U+W)/W is isomorphic to U/(U \cap W). Homework Equations Here the use of / denotes a quotient space. The Attempt at a Solution Not even sure where to begin.

Homework Statement Determine whether or not G is isomorphic to the product group HXK. G=ℂx H={unit circle} K={Positive real numbers} Homework Equations Let H and K be subroups of G, and let f:HXK→G be the multiplication map, defined by f(h,k)=hk. Its image is the set HK={hk...

Homework Statement Let G be a group of order p2, where p is a positive prime. Show that G is isomorphic to either Z/p2 or Z/p × Z/p. The Attempt at a Solution Am I completely wrong here or is this just the definition of a p-Sylow subgroup? what I mean is that if g is of order p2...

Information: The vector-space \mathcal{F}([0,\pi],\mathbb{R}) consists of all real functions on [0,\pi]. We let W be its subspace with the basis \mathcal{B} = {1,cost,cos(2t),cos(3t),...,cos(7t)}. T: W \rightarrow \mathbb{R} ^8 is the transformation where: T(h) = (h(t_1), h(t_2),...,h(t_8))...

There exists none. What's the easiest way to prove this? Can we state that all elements of ℤ are in ℤ[x] but not the other way around?

• \mathfrak{g} is the Lie algebra with basis vectors E,F,G such that the following relations for Lie brackets are satisfied: [E,F]=G,\;\;[E,G]=0,\;\;[F,G]=0. • \mathfrak{h} is the Lie algebra consisting of 3x3 matrices of the form \begin{bmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0...

Homework Statement Let \mathfrak{g} , \mathfrak{h} be Lie algebras over \mathbb{C}. (i) When is a mapping \varphi : \mathfrak{g} \to \mathfrak{h} a homomorphism? (ii) When are the Lie algebras \mathfrak{g} and \mathfrak{h} isomorphic? (iii) Let \mathfrak{g} be the Lie algebra with...

Homework Statement *Attached is the problem statement, along with a definition which is to be used (I feel like some of the definitions in this text are somewhat untraditional, so I am including this one for clarity). Edit* sorry, it's problem #31. Homework Equations The...

The necessary and sufficient condition for homomorphisim f of a group G into a group G' with kernel K to be isomorphism of G into G' is that k={e} ... THOUGH I AM ABLE TO PROVE THAT f IS ONE-ONE AND f IS HOMOMORPHISM (in converse part) BUT CAN'T GET ANY IDEA TO PROVE THAT f IS ONTO. PLEASE...

The dihedral group Dn of order 2n has a subgroup of rotations of order n and a subgroup of order 2. Explain why Dn cannot be isomorphic to the external direct product of two such groups. Please suggest how to go about it. If H denotes the subgroup of rotations and G denotes the subgroup of...

As the title suggest, I do not understand what the difference between isomorphism and equality is in terms of graph theory. I have tried searching the internet but the few definitions I could find for each did not really shed light on the difference. I know that an isomorphism is when there is a...

Is what I did all I need to do? Is there anything else I need to prove? http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110728_175901.jpg?t=1311905852 http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110728_175917.jpg?t=1311905865

Homework Statement Let (G,\cdot) be a group. Defining the new operation * such that a*b = b \cdot a it is pretty easy to show that (G,*) is a group. Show that this new group is isomorphic to the old one. Homework Equations The Attempt at a Solution I have been experimenting...

Homework Statement is Z252 X Z294 isomorphic to Z42 X Z1764? Explain. Homework Equations The Attempt at a Solution I checked that the highest order of the element in both group are 1764, but don't really know how to justify if there is an isomorphism...Can anyone give me some hints?

Just had an exam there, one of the questions was Partition the list of groups below into isomorphism classes 1.\mathbb{Z}_8 2.\mathbb{Z}_8^* (elements of Z_8 relatively prime to 8) 3.\mathbb{Z}_4 \times \mathbb{Z}_2 4.\mathbb{Z}_{14} \times \mathbb{Z}_5 5.\mathbb{Z}_{10} \times...

Hi, I've come across this result which says that if there are two isomorphic vector spaces with a transformation between them, then that transformation must be linear. Can anyone help me prove this? For instance, if I have a transformation T: Z -> Z where Z is the set of integers, T(z) =...

In the conditions where the second isomorphism theorem applies, one has H/HnK = HK/K so in particular, taking orders (in the finite case), one has the order formula |HK| = |H|*|K|/|H n K|. Does anyone know if this formula holds in general, or under lesser hypotheses? Thx.

(Apologies for ascii art math, I don't know latex. Also apologies if this is in the wrong forum.) Homework Statement Why, in this lemma, must there be a vector v in V? That is, why must V be nonempty? An isomorphism maps a zero vector to a zero vector. Where f:V->W is an isomorphism, fix any...

natural isomorphism from V to V** It is known that there is a natural isomorphism \epsilon \rightleftharpoons \omega^\epsilon from V to V**, where \omega: V \times V* \rightarrow R is a bilinear mapping. So given a certain \epsilon \in V, its image under the isomorphism is actually a set of...

3. Let R = a+b \sqrt{2} , a,b is integer and let R_{2} consist of all 2 x 2 matrices of the form [\begin{array}{cc} a & 2b \\ b & a \\ \end{array} }] Show that R is a subring of Z(integer) and R_{2} is a subring of M_{2} (Z). Also. Prove that the mapping from R to R_{2} is a isomorphism.

Im having difficulty understanding this satement - can someone please explain it to me... let M be the class of mobius transformations M is isomorphic to GL2/Diag isomorphic to SL2/Id, where GL2 is the group of non-degenerate matrices of size 2 x 2 with complex entries, SL2 = A in GL2 ...

Homework Statement Suppose H is a normal subgroup G and L is a subgroup of K. Then (G x K)/(H x L) is isomorphic to (G/H) x (K/L) Homework Equations The Attempt at a Solution I know that I have to use the First Isomorphism Theorem, but in order to do that I need some function phi...

Consider: \varphi:R\rightarrow S is a homomorphism. Also,\hat{\varphi}:\frac{R}{ker\varphi}\rightarrow \varphi(R). How can I show \hat{\varphi} is bijective? Most textbooks say it is obvious. I see surjectivity obvious but not injectivity. Could anyone provide a proof for injectivity?

Let G be a group, H a normal subgroup, N a normal subgroup, and H intersect N = {e}. Let H x N be the direct product of H and N. Prove that f: HxN->G given by f((h,n))=hn is an isomorphism from HxN to the subgroup HN of G. Hint: For all h in H and n in N, hn=nh.

Homework Statement Let t \in L(V,W). Prove that t is an isomorphism iff it carries a basis for V to a basis for W.Homework Equations L(V,W) is the set of all linear transformations from V to WThe Attempt at a Solution So I figured I would assume I have a transformation from a basis for V to a...

I was asked to decide if Z_9 and the direct sum of Z_3 and Z_3 are isomorphic. Do I check to see if they are 1-1 and onto?

Homework Statement Consider the groups Q+ and Q* (rational under addition and ration under multiplication). Prove that neither of these groups is finitely geneated by using the fact that there are infinitely many primes. And prove that Q+ is not isomorphic to Q*. 2. The attempt at a...

Homework Statement Show that R^x/<-1> is isomorphic to the group of positive real numbers under multiplication. Homework Equations The Attempt at a Solution I know I need to show we have a homomorphism, and is one - to one and onto in order to be isomorphic. I know all...

Homework Statement I need to prove that any isomorphism between two cyclic groups maps every generator to a generator. 2. The attempt at a solution Here what I have so far: Let G be a cyclic group with x as a generator and let G' be isomorphic to G. There is some isomorphism phi: G...

Homework Statement Determine whether R is isomorphic to S for each pair of rings given. If the two are isomorphic, find an explicit isomorphism (you do not need to show the formal proof). If not, explain why. Homework Equations R= 2x2 matrix, a 0, 0 b, for some integers a,b S= Z x Z...

I'm trying to understand the first isomorphism theorem for groups. Part of the examples given in the book is showing that Q[x]/(x^3-3) is isomorphic to {a+b*sqrt(3)} As I understand it, by finding a homomorphism from Q[x] to {a+b*sqrt(3)} in which the kernel is x^3-3, the two are...

here f:Z36 to Z4 x Z9 is group isomorphism given by f(n+36Z)= (n+4Z, n+9Z) then what is the inverse of f ?

Homework Statement Why does it make sense (when considering Z4)to form the factor group Z4 / (2Z4) where kZn = {0, k mod n, 2k mod n, ..., nk mod n}? I believe that this above factor group is isomorphic to Z2, but how can I prove this?

Why does it make sense ( when considering Z4)to form the factor group Z4 / (2Z4) where kZn = {0, k mod n, 2k mod n, ..., nk mod n}? I believe that this above factor group is isomorphic to Z2, but how can I prove this?

Homework Statement Is there an isomorphism between O(2n)\simeq SO(2n)\times \mathbb{Z}_2 O(2n+1)\simeq SO(2n+1)\times \mathbb{Z}_2 Homework Equations First isomorphism theorem The Attempt at a Solution I think, if I can show a homomorphism between SO(2n)\times\mathbb{Z}_2...

Homework Statement Let G be any group and let a be a fixed element of G. Define a function c_{a}:G-->G by c_{a}(x)=axa^{-1} for all x in G. Show that c is an isomorphism The Attempt at a Solution Need to show 1-1, onto and c(ab)=c(a)c(b) I guess my biggest problem is starting because I...

Let J be a set of all linear functions. Consider the set R^2 in the Euclidean plane. Define a binary operation * on R^2 in such a way that the two binary structures <J, +> and <R^2, *> will be isomorphic. Any thoughts? If something is not clear please ask. Thank you.

The poset on the set of order ideals of a poset p, denoted L(p), is a distributive lattice, and it is pretty clear why this is since the supremum of two order ideals and the infimum of 2 order ideals are just union and intersection respectively, and we know that union and intersection are...

How to prove that a group of order prime number is cyclic without using isomorphism/coset? Can i prove it using basic knowledge about group/subgroup/cyclic(basic)? I just learned basic and have not yet learned morphism/coset/index. Can you guys kindly give me some hints or just answer...

Homework Statement Let F be the set of all functions f mapping R into R that have derivatives of all orders. Determine whether p is an isomorphism of the first binary structure with the second. 1. <F, +> with <R, +> where p(f) = f'(0) 2. <F, +> with <F, +> where p(f)(x) = \int^{x}_{0}...

Why these two tensor products are isomorphic? Hom_{K}(V,K) \otimes Hom_{K}(V,K) and Hom_{K}(V \otimes V,K) where K is a field and V is a vector space over K.

Hi i just start learning algebra. Here are some definitions and examples given in Wikipedia: 1.An isomorphism is a bijective map f such that both f and its inverse f^{-1} are homomorphisms, i.e., structure-preserving mappings. 2.A homomorphism is a structure-preserving map between two algebraic...

What does "Up to Isomorphism" mean? I was reading the final chapter in Spivak's Calculus and it says: There is a complete ordered field and, up to isomorphism, only one complete ordered field. I know what an isomorphism is and what it means when things are isomorphic. But I don't know...

I have not seen why SU(2) and SO(3) groups are isomorphic?