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2. Prove that a group G is abelian if and only if the mapping f G→ G, given by f(x)=x-¹, is a homomorphism: 3. Show that a group G is abelian if and only if the mapping f: G→G, given by f(x)= x², is a homomorphism. 4. Find the kernel of each of the following homomorphism: (a) ƒ: Z→ Z„, given by ƒ(x)= X. (b) f: G-Z₂, where G is the quaternion group (see Problem 6, Section 1) and f(a)=Ō, ƒ (b) = ī.

2. Prove that a group G is abelian if and only if the mapping f G→ G, given by f(x)=x-¹, is a homomorphism: 3. Show that a group G is abelian if and only if the mapping f: G→G, given by f(x)= x², is a homomorphism. 4. Find the kernel of each of the following homomorphism: (a) ƒ: Z→ Z„, given by ƒ(x)= X. (b) f: G-Z₂, where G is the quaternion group (see Problem 6, Section 1) and f(a)=Ō, ƒ (b) = ī.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Prove that a group G is abelian if and only if the mapping f G-G, given by f(x)=x-¹, is a homomorphism: 3. Show that a group G is abelian if and only if the mapping f: G→G, given by f(x)=x², is a homomorphism. 4. Find the kernel of each of the following homomorphism: (a) f: Z→Z,, given by f(x)= x. (b) f: G-Z₂, where G is the quaternion group (see Problem 6, Section 1) and f(a)=Ō, ƒ (b) = ī. Show that there does not exist any nonzero homomorphism of
Transcribed Image Text:2. Prove that a group G is abelian if and only if the mapping f G-G, given by f(x)=x-¹, is a homomorphism: 3. Show that a group G is abelian if and only if the mapping f: G→G, given by f(x)=x², is a homomorphism. 4. Find the kernel of each of the following homomorphism: (a) f: Z→Z,, given by f(x)= x. (b) f: G-Z₂, where G is the quaternion group (see Problem 6, Section 1) and f(a)=Ō, ƒ (b) = ī. Show that there does not exist any nonzero homomorphism of
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