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. PragmaticYak

    Fixed point free automorphism of order 2

    I did not use the hint for this problem. Here is my attempt at a proof: Proof: Note first that ##σ(σ(x)) = x## for all ##x \in G##. Then ##σ^{-1}(σ(σ(x))) = σ(x) = σ^{-1}(x) = σ(x^{-1})##. Now consider ##σ(gh)## for ##g, h \in G##. We have that ##σ(gh) = σ((gh)^{-1}) = σ(h^{-1}g^{-1})##...
  2. AutGuy98

    MHB 4 part problem regarding homomorphisms

    Hey guys, I have some more problems that I need help with figuring out what to do. The first one is divided into 4 mini-problems, sub-sections, whatever you would like to call them. It asks: (a) Show that (Z/4Z,+) is not isomorphic to ((Z/2Z) x (Z/2Z),+). Find a homomorphism from (Z/4Z,+) to...
  3. A

    Show injectivity, surjectivity and kernel of groups

    Homework Statement I am translating so bear with me. We have two group homomorphisms: α : G → G' β : G' → G Let β(α(x)) = x ∀x ∈ G Show that 1)β is a surjection 2)α an injection 3) ker(β) = ker(α ο β) (Here ο is the composition of functions.) Homework Equations This is from a...
  4. T

    I Images of elements in a group homomorphism

    Why does the image of elements in a homomorphism depend on the image of 1? Why not the other generators?
  5. evinda

    MHB Number of Homomorphisms from $\mathbb{Z}_4$ to $S_4$: A Brief Exploration

    Hello! (Wave) We are given the groups $G_1=\mathbb{Z}_4$ and $G_2=S_4$. We consider the homomorphisms $f: G_1 \to G_2$. Let $k$ be the number from all of these $f$. What is $k \bmod{6}$ equal to ? How can we find the number of homomorphisms $f$? Could you give me a hint? (Thinking)
  6. PsychonautQQ

    I Understand homomorphisms from Z^a --> Z^b

    I want to understand all possible homomorphisms ##\alpha: Z^a -> Z^b## as well as understand what a matrix representation for an arbitrary one of these homomorphisms would look like. Furthermore, under what conditions does a homomorphism have a matrix representation? To begin, let...
  7. U

    Are these homomorphisms?

    Homework Statement Are these functions homomorphisms, determine the kernel and image, and identify the quotient group up to isomorphism? C^∗ is the group of non-zero complex numbers under multiplication, and C is the group of all complex numbers under addition. Homework Equations φ1 : C−→C...
  8. Math Amateur

    MHB Division Rings and RIng Homomorphisms .... A&W Corollary 2.4 ....

    I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Corollary 2.4 ... ... Corollary 2.4 and its proof read as follows: In the above proof of Corollary...
  9. Math Amateur

    I Division Rings & Ring Homomorphisms .... A&W Corollary 2.4 ...

    I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Corollary 2.4 ... ... Corollary 2.4 and its proof read as follows: In the above proof of...
  10. M

    MHB How many homomorphisms are there from $\mathbb{Z}_4$ to $S_4$?

    Hey! :i How many homomorphism $f:\mathbb{Z}_4\rightarrow S_4$ are there? Do we have to find how many permutations of $S_4$ have order that divides $4$ ? We have 1 identity (order 1), 6 transpositions (order 2), 3 products of two disjoint transpositions (order 2), 6 4-cycles (order 4). So...
  11. Math Amateur

    MHB Ring Homomorphisms from Z to Z .... Lovett, Ex. 1, Section 5.4 .... ....

    I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 5.4 Ring Homomorphisms ... I need some help with Exercise 1 of Section 5.4 ... ... ... Exercise 1 reads as follows:Relevant Definitions A ring homomorphism is defined by...
  12. Math Amateur

    Ring Homomorphisms from Z to Z .... Lovett, Ex. 1, Section 5.

    Homework Statement I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 5.4 Ring Homomorphisms ... I need some help with Exercise 1 of Section 5.4 ... ... ... Exercise 1 reads as follows: Homework Equations The relevant...
  13. Math Amateur

    MHB Basic Question on Ring Homomorphisms

    I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help to clarify a remark of B&K regarding ring homomorphisms from the zero or trivial ring ... The relevant text from B&K reads as follows...
  14. Math Amateur

    I Basic Question about a Ring Homomorphisms

    I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help to clarify a remark of B&K regarding ring homomorphisms from the zero or trivial ring ... The relevant text from B&K reads as follows: In the above text from B&K's book...
  15. Math Amateur

    MHB Ring Homomorphisms - Rotman - Theorem 3.33

    I am reading Joseph J.Rotman's book, A First Course in Abstract Algebra. I am currently focused on Section 3.4 Homomorphisms (of Rings) I need help with the proof of Theorem 3.33 ... Theorem 3.33 and the start of its proof reads as follows: https://www.physicsforums.com/attachments/4529 In...
  16. B

    Projection Functions and Homomorphisms

    Homework Statement Let ##G##, ##H##, and ##K## be groups with homomorphisms ##\sigma_1 : K \rightarrow G## and ##\sigma_2 : K \rightarrow H##. Does there exist a homomorphism ##f: K \rightarrow G \times H## such that ##\pi_G \circ f = \sigma_1## and ##\pi_H \circ f = \sigma_2##? Is this...
  17. B

    Homomorphisms and Calculating Elements

    Homework Statement Suppose that ##G## is a cyclic group with generator ##g##, that ##H## is some arbitrary group, and that ##\phi : G \rightarrow H## is a homomorphism. Show that knowing ##\phi (g)### let's you compute ##\phi(g_1)## ##\forall g_1 \in G## Homework Equations ##\phi(g^n) =...
  18. J

    Groups of homomorphisms of abelian groups

    Hello everybody! I've just started with studying group homorphisms and tensor products, so i am still not very sure if i undertstand the subject correct. I am stuck with a question and i would ask you for some help or hints how to proceed... What i have to do is to describe...
  19. TrickyDicky

    Possible title: When Does the Kernel of a Homomorphism Reduce to the Identity?

    I'm centering on lie group homomorphisms that are also covering maps from the universal covering group. So that if their kernel was just the identity they would be isomorphisms. Are there situations in which the kernel of such a homomorphism would reduce to the identity? I'm thinking of...
  20. Math Amateur

    MHB Quotient Modules and Module Homomorphisms - Cohn - Corollary 1.16

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics we find Corollary 1.16 on module homomorphisms and quotient modules. I need help with some aspects of the proof. Corollary 1.16 reads as follows: In the above text...
  21. Math Amateur

    MHB Quotient Modules and Homomorphisms

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics we find Theorem 1.15 on module homomorphisms and quotient modules. I need help with some aspects of the proof. Theorem 1.15 reads as follows: In the proof of the...
  22. W

    Bijective Homomorphisms and Isomorphisms

    Hi All, Let A,B be algebraic structures and let h A-->B be a bijective homomorphism. Is h an isomorphism? In topology, we have continuous bijections that are not homeomorphisms, (similar in Functional Analysis )so I wondered if the "same" was possible in Algebra. I assume if there is a...
  23. Math Amateur

    Exact Sequences - Lifting Homomorphisms - D&F Ch 10 - Theorem 28

    I am reading Dummit and Foote, Chapter 10, Section 10.5, Exact Sequences - Projective, Injective and Flat Modules. I need help with a minor step of D&F, Chapter 10, Theorem 28 on liftings of homomorphisms. In the proof of the first part of the theorem (see image below) D&F make the following...
  24. Math Amateur

    MHB Exact Sequences - Lifting Homomorphisms - D&F Ch 10 - Theorem 28

    I am reading Dummit and Foote, Chapter 10, Section 10.5, Exact Sequences - Projective, Injective and Flat Modules. I need help with a minor step of D&F, Chapter 10, Theorem 28 on liftings of homomorphisms. In the proof of the first part of the theorem (see image below) D&F make the following...
  25. Math Amateur

    MHB Exact Sequences - extending or lifting homomorphisms

    Dummit and Foote open their section (part of section 10.5) on projective modules as follows:D&F then deal with the issue of obtaining a homomorphism from D to M given a homomorphism from D to L and then move to the more problematic issue of obtaining a homomorphism from D to M given a...
  26. Math Amateur

    (Apparently) simple question rearding module homomorphisms

    I am reading Dummit and Foote Chapter 10: Introduction to Module Theory. I am having difficulty seeing exactly why a conclusion to Proposition 27 that D&F claim is "immediate": I hope someone can help. Proposition 27 and its proof read as follows: In the first line of the proof...
  27. Math Amateur

    MHB (Apparently) simple question rearding module homomorphisms

    I am reading Dummit and Foote Chapter 10: Introduction to Module Theory. I am having difficulty seeing exactly why a conclusion to Proposition 27 that D&F claim is "immediate": I hope someone can help. Proposition 27 and its proof read as follows...
  28. Math Amateur

    MHB Exact Sequences - Split Sequences and Splitting Homomorphisms

    I am reading Dummit and Foote Section 10.5 Exact Sequences - Projective, Injective and Flat Modules. I need some help in understanding D&F's proof of Proposition 25, Section 10.5 (page 384) concerning split sequences. Proposition 25 and its proof are as follows...
  29. K

    MHB Sizes of kernels of homomorphisms

    I have a problem that I have been stuck on for two hours. I would like to check if I have made any progress or I am just going in circles. **Problem: Let $\alpha:G \rightarrow H, \beta:H \rightarrow K$ be group homomorphisms. Which is larger, $\ker(\beta\alpha)$ or $\ker(\alpha)$?** **My...
  30. R

    Homomorphisms of Quaternion Group

    Homework Statement Let Q = {±1, ±i, ±j, ±k} be the quaternion group. Find all homomorphisms from Z2 to Q and from Z4 to Q. Are there any nontrivial homomorphisms from Z3 to Q? Then, find all subgroups of Q. Homework Equations The Attempt at a Solution I don't even know...
  31. Krizalid1

    MHB How to Prove Certain Properties of Homomorphisms and Ideals in Ring Theory?

    Let $\phi:R\to S$ be a homomorphism of rings. Let $I$ be an ideal of $R$ and $J$ be an ideal of $S.$ Prove that $\phi^{-1}(J)$ is an ideal of $R$ and $\ker(\phi)\subset\phi^{-1}(J).$ Also prove that $\phi(I)$ is not necessarily an ideal of $S.$
  32. M

    Homomorphisms of Languages

    Hey, I was thinking of this generalization of homomorphisms. You have a language L_1 = (A, B) where A is a set of symbols and B is a set of sequences of symbols in A. Given languages L_1 = (A, B) and L_2 = (C, D) a function f: A \rightarrow C is defined to be a homomorphism of languages if...
  33. ArcanaNoir

    All homomorphisms from Z_n to Z_m

    Homework Statement Describe all group homomorphisms from \mathbb{Z}_n to \mathbb{Z}_m . Homework Equations \mathbb{Z}_n = {[0],[1],\dots ,[n-1]} with addition. A homomorphism is an operation preserving map, ie \phi (a\ast b)=\phi (a) \# \phi (b) . One especially important...
  34. R

    Homomorphisms between two isomorphic rings ?

    Homework Statement True or False? Let R and S be two isomorphic commutative rings (S=/={0}). Then any ring homomorphism from R to S is an isomorphism. Homework Equations R being a commutative ring means it's an abelian group under addition, and has the following additional properties...
  35. C

    Homomorphisms with unknown groups

    Homework Statement 1)Let p,q be primes. Show that the only group homomorphism $$\phi: C_p \mapsto C_q$$ is the trivial one (i.e ## \phi (g) = e = e_H\,\forall\,g##) 2)Consider the function $$det: GL(n,k) \mapsto k^*.$$ Show that it is a group homomorphism and identify the kernel and...
  36. A

    A question about commutative rings and homomorphisms

    Let R be a commutative ring. Show that the function ε : R[x] → R, defined by \epsilon : a_0 + a_1x + a_2x +· · ·+a_n x^n \rightarrow a_0, is a homomorphism. Describe ker ε in terms of roots of polynomials. In order to show that it is a homomorphism, I need to show that ε(1)=1, right? But...
  37. A

    A question about ring homomorphisms

    Homework Statement If R is a domain with F=Frac(R), prove that Frac(R[x]) is isomorphic to F(x). Homework Equations The Attempt at a Solution Let \phi : Frac(R[x]) \rightarrow F(x) be a map sending (f(x),g(x)) to f(x)/g(x). We need to show that \phi is a ring homomorphism. Let f,g,h,k be in...
  38. A

    A question about ring homomorphisms

    I attached a page from my textbook, because there was something that I didn't understand. What I don't understand is in the proof it says let f(x) be...etc. but in the theorem, it says nothing about f(x). In other words, where in the thoerem does it say anything about f(x). Why are they...
  39. S

    Homomorphisms of Cyclic Groups

    So this is a pretty dumb question, but I'm just trying to understand homomorphisms of infinite cyclic groups. I understand intuitively why if we define the homomorphism p(a)=b, then this defines a unique homorphism. My question is why is it necessarily well-defined? I think I'm confused...
  40. J

    Homomorphisms of Polynomials Over Integral Domains

    Let A be an integral domain. If c ε A, let h: A[x] → A[x] be defined by h(a(x))=a(cx). Prove that h is an automorphism iff c is invertible. This one really had me stumped. I have a general idea of what the function is doing. Now, assuming that h is an automorphism, we want to show that...
  41. C

    MHB Homomorphisms and radical

    Let M be the set of 2x2 matrices defined by M = {a b 0 d} where a, b and d are complex. I've found a basis for M but need to know how to find the set of scalar homomorphisms of M from these. I have the basis as M_1 = {1 0 0 1} M_2 = {0 1 0 0} and M_3 = {0 0...
  42. M

    Find all ring homomorphisms from 3Z to Z?

    Homework Statement Find all ring homomorphisms from 3Z to Z, where 3Z are the integers that are of multiple 3. Homework Equations The Attempt at a Solution So 3Z is cyclic so σ(3) is sufficient to look. Now all of the other examples have finite groups, so |σ(a)| divides the |a|...
  43. W

    Galois Theory questions: Homomorphisms

    Homework Statement Let K = Q(2^(1/4)) a) Which of the morphisms from K to C are Q(2^1/2)-homomorphisms b) And which are K-homomorphisms? Homework Equations Theorem: There is a bijection between roots of minimal polynomial and number of homomorphisms Definition: A K-Homomorphism...
  44. W

    Galois Theory questions: Homomorphisms

    Let K = Q(2^(1/4)) a) Which of the morphisms from K to C are Q(2^1/2)-homomorphisms b) And which are K-homomorphisms? Attempt at a solution Ok, I don't really understand this very well but for a) I know that there are 4 homomorphisms, since the minimal polynomial over C has four...
  45. W

    How Many Homomorphisms Can Be Calculated Between Different Fields?

    Hi, I am trying to calculate the number of homomorphisms from one field to another: a) F2 ---> F3 b) Q[X]/(X7 - 3) ---> Q[X]/(X8 + 4X5 - 6X + 2) c) F7 [X] / (X2 + X - 1) ---> F7[X] / (X2 + 1) d) Q( 21/4 ) ---> C Attempt at a solution a) I'm pretty sure there are no homomorphisms between F2...
  46. L

    Order of Homomorphisms and Finite Elements

    Homework Statement Let ψ: G→H be a homomorphism and let g ε G have finite order. a) Show that the order of ψ(g) divides the order of gThe Attempt at a Solution I'm really lost here, but I'm guessing we can use the fact |ψ(g)| = {e,g...,g|g|-1} and ψ(g|g|-1) = ψ(g)ψ(g)ψ(g)ψ(g)ψ(g)... (|g|-1...
  47. S

    Describe all homomorphisms from \mathbb{Z},+ to \mathbb{Z},+

    Homework Statement Find all homomorphisms f: \mathbb{Z},+ \rightarrow \mathbb{Z},+. Determine which are injective, which are surjective, and which are isomorphisms. Note. I must prove everything.Homework Equations Notation. \mathbb{Z}n = \{ p : p = kn, \, \, \, \mathrm{k} \in \mathbb{Z} \}The...
  48. K

    R-module homomorphisms isomorphic to codomain

    Homework Statement Let R be a commutative ring, and M be an R-module. Show that \text{Hom}_{\text{R-mod}}(R,M) \cong M as R-modules, where the homomorphisms are R-module homomorphisms. The Attempt at a Solution This should hopefully be quick and easy. The most natural mapping to...
  49. M

    Ring homomorphisms of polynomial rings

    Homework Statement Let R be a commutative ring and let fa: R[x] -> R be evaluation at a \in R. If S: R[x] -> R is any ring homomorphism such that S(r) = r for all r\in R, show that S = fa for some a \in R. Homework Equations The Attempt at a Solution I don't get this at all...
  50. M

    Onto Homomorphisms of Rings

    Homework Statement f: R -> R1 is an onto ring homomorphism. Show that f[Z(R)] ⊆ Z(R1) Homework Equations The Attempt at a Solution I'm a little confused. So f is onto, then for all r' belonging to R1, we have f(r)=r' for some r in R. But if f is onto couldn't R1 has less...
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