11.1.1. Groups Homomorphisms **Question 1:**Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?\[\phi: \mathbf{M}_2(\mathbb{R}) \to \mathbb{R} \text{ by } \phi\left(\left(\begin{array}{cc} a & b \\ c & d \end{array}\right)\right) = b,\]where \(\mathbf{M}_2(\mathbb{R})\) is the additive group of \(2 \times 2\) matrices with entries in \(\mathbb{R}\).

Answered: Image /qna-images/answer/0fc241f3-3082-4585-bce2-e96dcde53817.jpgYes is an homomorphismThe kernal is

Answered: 1. Which of the following maps are…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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12.3.7. Our friend G is a group of order 63. (a) The group G is guaranteed to have subgroups of which sizes? Why? (b) Assume that H is a subgroup of G of order 21. Prove that H < G.
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(2) Suppose a group contains elements of order 1 through 9. What is the minimum possible order of the group? Part III
19. 20. 21. 22. 23. 24. 25. Consider the groups U(8) and Z4 c) d) e) f) Let H₁, H₂ and H, be abelian groups. Prove or disprove: H, xH₂x H, is an abelian group. List the elements of U(8) x Z4 Determine the identity element of this group. Determine all the elements which are of order two in this group. Determine all the elements of order 4 in this group.? Determine the subgroup generated by the element (7,1) Let K = {xe G|xg=gx Vge G} Show that K is a normal subgroup of G. Define f: R² R² by f(x,y) = (x + 2y, 0) h) i) a) b) Show that f is a homomorphism from into itself Find Ker(f) Let 0: - be defined by 0 la b C by σ(n) =i" a) b) Prove that is a homomorphism Determine Ker (0) Let G= - Verify that o is a homomorphism Find Ker(0) (a,b,c,de R ) Use the Fundamental homomorphism theorem to prove that 45
3.42. Let G be the set of all sequences of integers (a₁, a2, a3, . . .). 1. Show that G is a group under (a₁, a2, ...) + (b₁, b₂, ...) = (a₁ + b₁, a₂ + b₂,...). 2. Let H be the set of all elements (a₁, a2, ...) of G such that only finitely many a are different from 0 (and (0, 0, 0, ...) E H). Show that H is a subgroup of G.
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1. Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel? = b, where M₂ (R) is the additive group of 2 x 2 matrices with entries in R. 4: M₂ (R) → R by ø ((ab) φ: C