**Gaussian Integers and Group Homomorphism**Let $ G = \mathbb{Z}[i] = \{a + bi \mid a, b \in \mathbb{Z}\} $ be the Gaussian integers, which form a group under addition. Let $ \gamma \in \mathbb{R} $, and define a function $ \varphi : G \to \mathbb{R} $ by \[\varphi(a + bi) = a + \gamma b.\]Prove that $ \varphi $ is a group homomorphism. Furthermore, show that $ \varphi $ is injective if and only if $ \gamma $ is irrational.

Answered: Image /qna-images/answer/aaf33018-f748-4a2b-94a8-ecddf287115d.jpg

Let G = Z[i] = {a+bi | a, b € Z} be the Gaussian integers, which form a group under addition. Let y e R, and define a function p: G + R by p(a+bi) = a+yb. Prove that p is a group homomorphism. Furthermore, show that o is injective if and only if y is irrational.

Let G = Z[i] = {a+bi | a, b € Z} be the Gaussian integers, which form a group under addition. Let y e R, and define a function p: G + R by p(a+bi) = a+yb. Prove that p is a group homomorphism. Furthermore, show that o is injective if and only if y is irrational.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Suppose F1 and F2 are distinct reflections in a dihedral group Dn.Prove that F1F2 ≠ R0.

A: Given: F1 and F2 are distinct reflections in a dihedral group Dn. To Prove: F1F2≠R0.

Q: Define f: G→ G', where f(x) = (3x for every x € G. the f homorphism. Determine the f

Q: Let a be an element of a group G such that o(a) = r. Let m be a positive integer such that (m,r) =…

A: a∈GOa=rgcdm,r=1 We have to prove Oam=r

Q: Let C(x) represent the centralizes of a. If x is an element of group G with the order of x be 5 (|x|…

Q: Let G and H be groups and consider the function f: G → G × H defined by f(g) = (g, eH ) where eH is…

Q: Let H = {z E C* | Izl = 1}. Prove that C*/H is isomorphic to R+, the group of positive real numbers…

Q: Use properties of the linear transformation T(x, y) = (3x − 2y, x + y) to determine if the matrix…

Q: Let A = {fm,b : R → R | m ± 0 and fm,b(x) : = mx + b, m, b e R} be the group of affine m b functions…

A: We are given two groups. A={fm,b:ℝ→ℝ|m≠0 and fm,b(x)=mx+b,m,b∈ℝ} B=mb01 | m,b∈ℝ,m≠0. We need to…

Q: Consider the three basic vectors: (2,1,0,-1),(-1,1,1,1),(2,7,4,1) a. give two vectors that are in…

A: Given the three basic vectors: (2,1,0,-1),(-1,1,1,1),and (2,7,4,1) We have…

Q: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: x and y are two vertices of a Cayley digraph. ai’s and bj’s are generators of the Cayley digraph

Q: There is a streetlight every 120 feet between the library and the post office. There is a…

A: There is a streetlight every feet between the library and the post office.There is a streetlight in…

Q: Let Φ be an automorphism of C*, the group of nonzero complexnumbers under multiplcation. Determine…

Q: Let a and b be elements of a group. If |a| and |b| are relatively prime, show that intersects =…

A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…

Q: Let a and b belong to a group. If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: 5 Let G be the set of all functions f: R → R of the form f(x) = ax + b where a and b are real…

A: Introduction: Function composition is an operation where two functions are combined to form a new…

Q: Write Q(sqrt2) for the set described in exercise 3.1 (iii). given a non zero element a + bsqrt(2) of…

A: It refers to the set Q(2), which is the set of all numbers of the form a+b2, where a and b are…

Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

Q: Let G = {a +b 2 ∈ ℝ │ a, b ∈ ℚ }. Prove that the nonzero elements of G form a group under…

A: Note : In the question, it has to be a + b2 ∈ ℝ instead of a + b 2 ∈ ℝ. So, we are solving the…

Question

**Gaussian Integers and Group Homomorphism**

Let $ G = \mathbb{Z}[i] = \{a + bi \mid a, b \in \mathbb{Z}\} $ be the Gaussian integers, which form a group under addition. Let $ \gamma \in \mathbb{R} $, and define a function $ \varphi : G \to \mathbb{R} $ by

\[
\varphi(a + bi) = a + \gamma b.
\]

Prove that $ \varphi $ is a group homomorphism. Furthermore, show that $ \varphi $ is injective if and only if $ \gamma $ is irrational.

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

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps with 2 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

Find an isomorphism & from the multiplicative group G of nonzero complex numbers to the multiplicative group a -b H a,b E R and a² + b² # 0 a and prove that p(xy) = 4(x)4(y).
Show that the determinant gives a group homomorphism det: GL(n,R)->R*, where R*=R-{0} under multiplication.
Let : G H be a group homomorphism. Prove that Kery = {e} if and only if is injective.
Determine whether an onto homomorphism between the groups D6 and D3 + Z2 exists.
Let G=S4, the symetric group of on 4 elem4nts. h) Show that A4 is nolmal by finding a homomorphism whose kernel is A4.
If R2 is the plane considered as an (additive) abelian group, show that any line L through the L in R2. Provide a origin is a subgroup. Now, if BE R2, then 8+ L represents the left coset geometric description of this coset and explain how you arrived at this description.
Use the method of undetermined coefficients to determine the form of a particular solution for the given equation. y'"' + 10y" – 11y=xe* + 4x What is the form of the particular solution with undetermined coefficients? Yp(x) = ] (Do not use d, D, e, E, i, or I as arbitrary constants since these letters already have defined meanings.)
Please don't provide handwritten solution ....
Prove that two cycles (a1,a2,...ak), (b1,b2,...bm) are disjoint cycles.
Let G = { a+bi ∈ ℂ : a and b are rational numbers, not both zero } . Prove that G with the operation of multiplication is a subgroup of ℂ×.
5. G (a + b.2: a eZ and b e Z) is a group under addition in the reals. Define o : G -→G for all a + b.2 e G, as 0(a+ b.2) = a- b./2. Prove that o is an automorphism.