Homomorphisms Definition and 96 Threads

Describe explicitly all homomorphisms h: C_6 ----> Aut(C_n) The question asks when n=12,16 I was wondering if someone could explain how to do this? I've looked through the notes but struggling a tad I think I could do this if it said for instance h: C_6 ----> C_n but Aut(C_12) = C_2 x C_2...

Homework Statement Consider the set Hom of homomorphisms from \mathbb{Z}^n (the n-dimensional integer lattice) to a group G . Also let S = \left\{ \, ( g_1, g_2, \dots, g_n ) \, | \, g_i g_k = g_k g_i, \text{where} \, 0 < i,k \leq n, g_i \in G \right\}, the set of n-tuples from G...

Homework Statement 1. Let G and H be finite groups and let a: G → H be a group homomorphism. Show that if |G| is a prime, then a is either one-to-one or the trivial homomorphism. 2. Let G and H be finite groups and let a : G → H be a group homomorphism. Show that if |H| is a prime, then a...

Homework Statement For groups G1 and G2, let p1 : G1 × G2 → G1 be defined by p1((g1, g2)) = g1 and let p2 : G1 × G2 → G2 be defined by p2((g1, g2)) = g2. Show that p1 and p2 are group homomorphisms and determine the kernel and image of each. Homework Equations The Attempt...

Homework Statement Show that the function i : Z12 → Z12 defined by i([a]) = 3[a] for all [a] ∈ Z12 is a group homomorphism and determine the kernel and image. Homework Equations The Attempt at a Solution Well, I started by computing i([a]i([b]) =3[a]3[b] =9[ab] It should equal...

If you have the dihedral group D4 and the symmetric group S8 how do you come up with a 1-to-1 group homomorphism from D4 to S8. I know what the multiplication table looks like. How can I use that to create the homomorphism? Let R1,. . . ., R4 represent the rotation symmetry. Let u1, u2...

Homework Statement Consider the group D5, the set of all twists and flips which we can perform on a regular pentagonal plate to pass through a fixed regular pentagonal hole under composition. a. Find all subgroups of D5 of order 2, if order 3, and of order 5. b. Find all homomorphisms...

Suppose \mathfrak{g} and \mathfrak{h} are some Lie algebras, and G=\exp(\mathfrak{g}) and H=\exp(\mathfrak{h}) are Lie groups. If \phi:\mathfrak{g}\to\mathfrak{h} is a Lie algebra homomorphism, and if \Phi is defined as follows: \Phi:G\to H,\quad \Phi(\exp(A))=\exp(\phi(A))...

Should be simple, but can't figure out :) Why is that , for a field K, the linear independence of field homomorphisms g1, ..., gn: K -> K equivalent to the existence of elements a1, ..., an \in K such that the determinant det| gi(aj)| != 0 (...so, just like in a case of linear...

1. Suppose that H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of G of index 4 and that G/(H intersect K) is not cyclic. 2. Homework Equations - the back of my book says to use the Second Isomorphism Theorem for the first part which is... If K...

Homomorphisms as "structure-preserving" maps A function f between groups is said to be a homomorphism if it "preserves" the product in the sense that f(xy)=f(x)f(y). A function f between fields is said to be a homomorphism if it "preserves" both addition and multiplication in the sense that...

Homework Statement Let R and S be rings. Show that \pi:RxS->R given by \pi(r,s)=r is a surjective homomorphism whose kernel is isomorphic to S. Homework Equations The Attempt at a Solution To show that \pi is a homomorphism map, I need to show that it's closed under addition and...

Homework Statement Assuming the mapping Z --> F defined by n --> n * 1F = 1F + ... + 1F (n times) is a ring homomorphism, show that its kernel is of the form pZ, for some prime number p. Therefore infer that F contains a copy of the finite field Z/pZ. Also prove now that F is a finite...

Okay, so I'm trying to finish of a problem on integral closure and I am rather unsure if the following fact is true: If L embeds into an algebraically closed field K and F is an algebraic extension of L, then it is possible to extend the embedding of L to F into K. Now the case where F...

Homework Statement Given two fields F,E with different characteristic. Prove or disprove the following statement: "Field homomorphisms between fields of different characteristic cannot exist" Homework Equations T : F1 --> F2 is a field homomorphism if 1) T(a+b) = T(a) + T(b) 2) T(ab)...

Abstract algebra question. Given the short exact sequence $ 1 \longrightarrow N \longrightarrow^{\phi} G \longrightarrow^{\psi} H \longrightarrow 1 $ I need to show that given a mapping $ j: H \longrightarrow G, and $ \psi \circ j = Id_h $ (the identity on H), then $ G \cong N \times H. (The...

Homework Statement Show that there are exactly two homomorphisms f:C_(6) --> C_(4) Homework Equations Theorem. let f: G -> G1 and h: G -> G1 be homomorphisms and assume that G=<X> is generaed by a subset X. Then f = h if and only if f(x) = h(x) for all x in X. The Attempt at a...

I've started self-studying algebra. So I want to err on the side of getting guidance so I don't get off on the wrong track. This is problem 2.4.4 in Artin. Describe all homomorphisms from Z+ to Z+ (all integers under addition). Determine if they are injective, surjective, or...

Homework Statement Find all ring homomorphisms \phi: Z \rightarrow Z \phi: Z2 \rightarrow Z6 \phi: Z6 \rightarrow Z2 Homework Equations A function \phi: R \rightarrow S is called a ring homomorphism if for all a,b\inR, \phi(a+b) = \phi(a) + \phi(b) \phi(ab) = \phi(a)\phi(b) \phi(1R)...

Homework Statement Let A, B be groups and A' and B' be normal subgroups of A and B respectively. Let f: A --> B be a homomorphism with f(A') being a subgroup of B'. There is a well-defined homomorphism g: A/A' -----> B/B' defined by g: aA' ---> f(a)B' Find an example in which f is...

Homework Statement Prove that if f: G \to H and g: H \to K are homomorphisms, then so is g \circ f: G \to K. 2. The attempt at a solution Since f is a homomorphism (G, * ) and (H, \circ) are groups and f(a*b)= f(a) \circ f(b), \forall a,b \in G. Likewise, (K, +) is a group and g(f(a)...

Homework Statement How many homomorphism are there of a free abelian group of rank 2 into a) Z_6 and b) S_3.Homework Equations The Attempt at a Solution Since the images of the generators completely determine a homomorphism, the upper bound for both is 36. Now a free abelian group of rank 2 is...

Homework Statement Show that the number of group homomorphisms from Zn to Zm is equal to gcd(n,m). my attempt: any hom from Zn to Zm must be f([x])=[kx] where k is a common factor of n and m. I can only get this far... any help is appreciated.

hi a little help would be kindly appreciated here guys. any suggestions on how to go about doing these? INFORMATION ----------------------- if K,Q are groups \varphi : Q \rightarrow Aut(K) is a homomorphism the semi direct product K \rtimes_{\varphi} Q is defined as follows. (i) as...

Homework Statement (i) Every group-theoretic relation p=q satisfied by (a,b,c) in G a group is also satisfied by (x,y,z) in F a group. (ii) There exists a homomorphism between G and F a->x b->y c->z. Problem: Show by example (i) can hold and (ii) cannot. Show (i) can hold and (ii)...

Homework Statement Find all homomorphisms from Z/<5> into Z/<5>. The Attempt at a Solution Is this a brute force question where we consider all the possibilities for the function? i.e f(0)=0,1,2,3,4 But that would still be combinatorially difficult.

Homework Statement I want to show that the homomorphism phi:A(X)->k+k given by taking f(x_1,...,x_n)-> (f(P_1),f(P_2)) is surjective. That is, given any (a,b) in k^2 (with addition and multiplication componentwise) I want to find a polynomial that has the property that f(P_1)=a and f(P_2)=b...

Homework Statement What is the difference between multiplicative functions and homomorphisms? Homework Equations The Attempt at a Solution

[SOLVED] Group Homomorphisms Thanks in advance for any help on this problem I can't even pretend that I know how to go about this question. I'm quite lost. Though thus far studying modern algebra hasn't been too difficult (knock on wood) and I've been understanding I'm struggling with this...

Homework Statement Let R* be the group of nonzero real numbersunder multiplications. Then the determinant mapping A->det A is a homomorphism from GL(2,R) to R* . The kernel of the determinant mapping is SL(2,R). Homework Equations The Attempt at a Solution I know...

Hello there, So I've noticed that at least out of the sources I've read, none of the point out the connection between additivity (a key operation that is emphasized on many texts) and homomorphisms. After all, a homomorphic function is merely a function wherein f(xy) = f(x)f(y). So a...

Homework Statement A function f:R-->R^x is a homomorphism iff f(x+y) = f(x) + f(y) for all x,y in R Homework Equations I don't know what group R^x is. I can only assume it means Reals under multiplication . Would that mean that f(x+y) = f(x)f(y)? How does the function work? Since 5...

I need to find all the ring homomorphisms of f:Z[sqrt(2)]->Z 7 basically I don't even know where to start. any suggestions would be great

Homework Statement Letv_1,v_2,...v_n be a basis of V and let w_1,w_2,...w_n be any n elements in V. Define T on V by (\lambda_1 v_1+\lambda_2 v_1+...+\lambda_n v_n)T=\lambda_1w_1+...\lambda_n w_n. a)Show that R is a homomorphism of V into itself. b)When is T an isomorphism? Homework...

Describe al group homomorphisms \phi : C_4 --> C_6 The book I study from seems to pass over Group Homomorphisms very fast. So I decided to look at Artin's to help and it uses the same definition. So I think I am just not digesting something I should be. I know it's defined as \phi (a*b)...

I've already completed 1), but it's necessary for one to know it for question 2). I'm pretty sure that I've found my homomorphism in 2, but I don't know whether or not is unique. How do I show a homomorphism is unique in this case? Problem 1: Let R be a commutative unital ring, and let S be a...

I just need confirmation. I have a problem in my algebra class that says: Prove that there are no ring homomorphisms from Z5 to Z7. I have the following definition of ring homomorphism: Let R and S be rings. A function R to S is a ring homomorphism if the following holds: f(1R)=1S...

How many homomorphisms are there from S_5 to \mathbb{Z}_5? Well there is at least one, the trivial homomorphism, ie: every element of S_5 gets mapped to 0. I have a feeling that this is the only homomorphism but am having trouble proving that no other homomorphism could exist. Any...

How does one compute the number of ring homomorphisms from \mathbb{Z}_2^n to \mathbb{Z}_2^m? Or, likewise, the number of linear mappings on those two vector spaces?

Is there a way to systematic way of counting the number of distinct homomorphisms from one ring to another?

i was just wondering if someone (matt) could give me a better idea of what the difference is between the two...thanks

How do you find all the homomorphisms from Z12 to Z6? and classify them by their kernals?

If you can help, that would be great. Let R be a commutative ring, and A,B,M be R-modules. Prove: a) HomR(A x B, M) is isomorphic to HomR(A, M) x HomR(B, M) b) HomR(M, A x B) is isomorphic to HomR(M, A) x HomR(M, B)

Consider the cyclic group G={a,a^2,a^3,...a^12=u} and its subgroup G`={a^2,a^4,...,a^12}. My book says that the mapping a^n ---> a^2n is an homomorphism of G onto G` (this seems true) and that X: a^n ---> a^n is homomorphism of G` onto G (this seems to be false to me, a misprint) A...

I'm trying to figure out all of the homomorphisms from Z onto Z mod 12. I can't figure out the trick - how am I possibly going to find ALL of the homomorphisms? Thanks - Colleen

I'm having a very tough time understanding homomorphisms and ideals, probably because I'm very fuzzy with the concept of rings. I'm stuck on the following problem: Find all the ideals in the following rings: 1. Z 2. Z[7] (Z subscript 7, equivalence classes of 7 I'm guessing) 3. Z[6] 4...