1-Which of the following is a homomorphism?f:Z Z, f(r) = x + 2a)030b) Of: Z12 Z30, ƒ(x) = 20x)of: Z5 → Z30, ƒ(x) = 6xd) O f: R R, f(x) = x²%3D

Consider a function .Then f is a homomorphism if and only if . .

Answered: 1- Which of the following is a…

1- Which of the following is a homomorphism? a)O f:Z→Z, f(x) = x+2 %3D b) O f: Z12 → Z30, ƒ(x) = 20x %3D 0of: Z5 → Z30, f(x) = 6x %3D d) f:R - R, f(x) = x²

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.6: Homomorphisms

Problem 3TFE: Label each of the following statements as either true or false. Every endomorphism is an...

See similar textbooks

Related questions

Q: 0:C' →C',o(x) = x*ANYWAY. O SOW THAT THERE IS A HOmomoRPHISm AND KER(Ơ) DETERMINE

A: Group Homomorphism: A Homomorphism ϕ from a group G, ∘ to a group G¯, ⊛ is a mapping from Ginto G…

Q: 3.A) In(i+x) 1 lim Prove that %3D マー0

A: According to the guidelines only one question will be answered. Please upload others separately.…

Q: 7. Compute the Jacobian of the following transformation. x=u²+uv y=3uv².

A: The given data is: x=u2+uvy=3uv2 To find the Jacobian transformation.

Q: 02 Define a Map L: M232M.3 by 1 3. LHA)-1 A, Is L a L2 lincar transformotion? Pove yovy AssertionB

Q: 2.4 (Ross, 2003, page 593) Show that the relationship $ (x) = x;¢(1;, x) + (1 – x;)ø(0¡, x) is true…

Q: Show that under the transformation: k= the equally spaced z-values, z : 1o, To +h, xo +2h, .…

A: Just substitute the given values of x and proof will follow.

Q: 1. For every element x in A, there is an element y in B such that (x, y) E F.

Q: 1. Suppose that f(2) is analytic on the set A = {z :1< |z| < 2}, and let f(2) = E apzk be its…

Q: List all the homomorphisms from D5 to Z2 + Z2.

A: There are 4 homomorphism from D5 to Z2+Z2

Q: 4. Show that in the following sense: for any ɛ > 0, there exists Ro >0 so that for any R> Ro, R rp

Q: 1. To show that the transform x - Ax = (x1- 2.x2 + x3,3x1+ x2,2x2 – x3), = (r1,#2, 13) E R³ is…

Q: If the Wronskian of the DE : x'y" + xp(x)y' + 3y 0 is et Then p(x) %3D -2x O

Q: 3. (a) Suppose N : V → V and N' : V → V are nilpotent of index r and s respectively and NN' = N'N.…

Q: Z6 is isomorphic to U(7) by giving an isomorphism φ from Z6 to U(7).

Q: (5,00 5 - Consider the function 0: (R {0}, )→ (R {0}, - ). Then, which of the followings is not a…

A: As per question asked we firstly define the property of homomorphism i.e. if x1, x2 belongs to x…

Q: 3 Let f: (A, SA)(8, 5,) be an order isomorphism and, Sa={xEA : x<ga}+ and Spa) -ly e8: f(5.)=Spa)…

A: Given: f: (A,⩽A) → (B,⩽B) is an order isomorphism Sa={x∈A: x<Aa}≠ϕ Sf(a)={y∈B: y<Bf(a)} To…

Q: Let T : V → U be a homomorphism, then Ker T = { 0 } iff T is one-one

A: A linear transformation is said to be injective or one to one if provided that for all v1 and v2 in…

Q: 1. Define f : (- o, 2) → (- 0, 1) by f(x) = . Show that f is bijective. x+1 *-2

Q: (8) Let f(x) ∈ Z[x] be an irreducible polynomial of degree 4 such that its Galois group over Q is…

A: Given, Letf(x)∈Z[x] be an irreducible polynomial of degree 4 such that its Galois group over Q is…

Q: Prove that d24 and s4 are not isomorphic. List elements and there order.

Q: 3.2 Consider R2 with (, ·) defined for all x and y in R? as х and 2 (x, y) := x |1 =:A Is (:, :) an…

A: The objective of the question check for inner product.

Q: — 2х21 Show that T: R² → R² defined by T()= - 15.) is an X2 isomorphism. X1

A: T is a n isomorphism

Q: 1. For every element x in A, there is an element y in B such that (x, y)EF

A: Remark: Let A and B be two non-empty sets. A relation R from A into B is subset of A*B. A function f…

Q: Consider X = [1,2]? with the relation (x1, y1)~(x2,y2) if x1y2 = X2y1. Also define g: X [1,2] by the…

Q: Show that the mapping :S3 S3 given by p(x) = xis not a homomorphism.

A: Solution is below.

Q: 32. Prove statement d of Theorem 3.9: If G is abelian, (xy)" = x"y" for all integers n. bns liD to…

A: I have proved using mathematical induction

Q: 15. Let x = (x1, x2). Using the feature mapping show that *(2,3)·¤((4,4) –-(2,3)·(4,4)*

A: Given that for x=x1,x2, the feature mapping is…

Q: 3. In each of the following examples, determine (with careful arguments) the bound- ary of A in X,…

A: Given X = R and A = (-1,1]

Q: Show that Z4 is not isomorphic to U(12) by comparing the orders of elements.

A: z4=0,1,2,3 is group w.r to +order of elementsoa=n⇒an=e where n is least positive…

Q: 1. Let O1 and O2 be topologies on X. (1) Show that the identity map idX : (X, O1) → (X, O2) is…

Q: Determine whether V and W are isomorphic. If they are, give an explicit isomorphism T: V--->W V…

A: Given, V = C, W = R2, and a map T : V ---> W. we have to show that whether V and W are…

Q: Find the Wronskian of {e¯*, xe-x, x²e¯x}.

A: Given that, The Wronskian of two differential function f and g is f (g') - g (f'). If f1, f2, ...,…

Q: Let G = Z, O Z, H = {(0, 0), (2, 0), (0, 2), (2, 2)}, and K = ((1, 2)). Is GIH isomorphic to Z, or…

Q: Let Qc(x) = x2 + c. Prove that if c 1 such that Qc is topologically conjugate to Fµ(x) = µx(1 − x)…

Q: n >= 1 , for every D € Dn as defined , D is an alternating multilinear map such that

Q: Let z = x + yi, z, = 1- 0.5i, and z, + z, = 3-1.5i. Find In(z,) - In(z2) C) In(50) + 2nin A) In(25)…

Q: Suppose there exist transposition f_i , 1<=I<=t such that

Q: If Rolle;s theorem applies on f(x) = 9 – (x – 3)², there exists a 'c' in [0, 6], Which satisfies the…

Q: 3. Let A : - ® be given by T2 In A(x1,..., Tn,.) = %3D 2 22 a) Prove that A is compact. b) Find the…

A: Given linear operator A:l2→l2 is defined by: Ax1,x2,…,xn,…=0,x12,x222,…,xn2n,…. (a). We Know that…

Q: Suppose m, n ∈ Z + are relatively prime. The function f : Zmn → Zm × Zn defined by f([Z]mn) =…

Q: 6. If X has full rank, so that P = X(X'X)-1X', prove that C(P) = C(X).

A: The rank of a matrix denotes the number of independent rows or the number of independent columns in…

Q: 1. Let 1, y € R" be given. (a) If ||z|| = ||y|| = 1 and z-y = 1, show that z = y. (b) If ||z|| ≤ 1…

A: \documentclass{article}\usepackage{amsmath,amssymb}\begin{document} (a) Norm x=Norm y=1\\ Let angle…

Q: In Z[x], let I = { f(x) E Z[x]: f(0) is an even integer). 1) Prove that I is an ideal of Z[x]. 2)…

Q: If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an…

A: No it doesn't follow the statement

Q: 4) Let m,n, r E Z with m # 0 + n. Show that gcd(m, n) divides r if and only if there exist a, b E Z…

A: Let gcd(m,n) divides r. Then we have to prove that, a·m+b·n=r Since, gcd(m,n) divides r than by…

Q: 3. In each of the following examples, determine (with careful arguments) the bound- ary of A in X,…

Q: (2) Suppose UC R" and V C R" are open subsets, and we have C maps U V 4U such gof is the identity of…

Q: (18) (a) Prove that R* is isomorphic to R>0 x Z2. (b) Is Q* is isomorphic to Qo x Z2? Prove your…

A: a) To prove :- ℝ*is isomorphic to ℝ>0×ℤ2. Proof :- Define a map ϕ : ℝ>0 ×ℤ2→ℝ* by…

Q: 4. Let w = e, and let E = Q(w). (a) Verify that [E : Q] = 10. (b) Determine |Gal(E/Q)|, the number…

A: Since you have asked multiple questions, we can solve first question for you. If you want other…

Question

1-
Which of the following is a homomorphism?
f:Z Z, f(r) = x + 2
a)
030
b) Of: Z12 Z30, ƒ(x) = 20x
)of: Z5 → Z30, ƒ(x) = 6x
d) O f: R R, f(x) = x²
%3D

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step by step

Solved in 5 steps with 33 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Groups$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Let be as described in the proof of Theorem. Give a specific example of a positive element of .
Prove that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.
A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.
Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
Prove that if f is a permutation on A, then (f1)1=f.
Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an isomorphism from G to G that is different from the one given in Example 5 of this section. Example 5 Consider G=1,i,1,i under multiplication and G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. In order to define a mapping :G4 that is an isomorphism, one requirement is that must map the identity element 1 of G to the identity element [ 0 ] of 4 (part a of Theorem 3.30). Thus (1)=[ 0 ]. Another requirement is that inverses must map onto inverses (part b of Theorem 3.30). That is, if we take (i)=[ 1 ] then (i1)=((i))1=[ 1 ] Or (i)=[ 3 ] The remaining elements 1 in G and [ 2 ] in 4 are their own inverses, so we take (1)=[ 2 ]. Thus the mapping :G4 defined by (1)=[ 0 ], (i)=[ 1 ], (1)=[ 2 ], (i)=[ 3 ]
23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.
6. Prove that if is a permutation on , then is a permutation on .
Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.
Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.