What is Homomorphisms: Definition and 96 Discussions

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).Homomorphisms of vector spaces are also called linear maps, and their study is the object of linear algebra.
The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory.
A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of those can be defined in a way that may be generalized to any class of morphisms.

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  1. M

    Exploring Homomorphisms and Automorphisms for Cyclic Groups

    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...
  2. D

    Mapping Homomorphisms to Commutative n-Tuples: A Bijection

    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...
  3. K

    Homomorphisms, finite groups, and primes

    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...
  4. K

    Homomorphisms and kernels

    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...
  5. K

    Homomorphisms and kernels,images

    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...
  6. Demon117

    Homomorphism Construction Using Symmetry Groups

    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...
  7. N

    Modern Algebra: Homomorphisms

    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...
  8. J

    Generating group homomorphisms between Lie groups

    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))...
  9. N

    Linearly independent field homomorphisms.

    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...
  10. E

    Abstract Algebra- homomorphisms and Isomorphisms, proving not cyclic

    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...
  11. Fredrik

    Homomorphisms as structure-preserving maps

    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...
  12. I

    Quotient Rings and Homomorphisms

    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...
  13. C

    Finite Fields and ring homomorphisms HELP

    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...
  14. S

    Homomorphisms into an Algebraically Closed Field

    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...
  15. I

    Do field homomorphisms preserve characteristic

    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)...
  16. M

    Short exact sequences and group homomorphisms

    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...
  17. H

    Are there any homomorphisms from C_6 to C_4?

    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...
  18. R

    What Are All the Homomorphisms from Z to Z?

    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...
  19. P

    Finding Ring Homomorphisms

    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)...
  20. P

    Needs a counterexample for homomorphisms

    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...
  21. F

    Composition of homomorphisms is a homomorphism

    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)...
  22. E

    Homomorphisms from a free abelian group

    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...
  23. H

    Number of group homomorphisms from Ｚ

    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.
  24. M

    Group Homomorphisms: Verifying Group Property & Finding Inverse

    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...
  25. M

    Solve Group Homomorphisms: Show (i) & (ii) Can Hold But Not Be Unique

    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)...
  26. M

    Group Homomorphisms from Z/<5> to Z/<5>: A Brute Force Approach

    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.
  27. Z

    Surjective Homomorphisms of Coordinate Rings

    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...
  28. E

    Multiplicative functions and homomorphisms

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

    Solving Group Homomorphisms Problem

    [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...
  30. B

    What is the kernel of the determinant mapping in GL(2,R)?

    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...
  31. Simfish

    Multiplicativity/Additivity and homomorphisms

    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...
  32. L

    Reals under multiplication homomorphisms

    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...
  33. A

    Find All Ring Homomorphisms f: Z[√2] → Z 7

    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
  34. happyg1

    Homomorphisms and isomorphisms

    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...
  35. G

    Group homomorphisms between cyclic groups

    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)...
  36. C

    Unital rings, homomorphisms, etc

    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...
  37. J

    There are no ring homomorphisms from Z5 to Z7

    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...
  38. J

    How many homomorphisms are there

    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...
  39. T

    How does one compute the number of ring homomorphisms

    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?
  40. T

    Counting Homomorphisms: A Systematic Approach

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

    Isomorphisms and homomorphisms

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

    Find all the homomorphisms from Z12 to Z6?

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

    Proving Module Homomorphisms: A x B to M &amp; M to A x B

    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)
  44. M

    Can a subgroup be mapped onto its parent group as a homomorphism?

    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...
  45. C

    Find ALL of the homomorphisms?

    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
  46. N

    What are the ideals in various rings?

    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...
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