4. Suppose (G₁, *) and (G₂, 0) are groups, let i and J denote the respective identities of G₁ and G2, let : G₁ G₂ be a group isomorphism, and suppose z E G₁. (a) Prove that if a has finite order, then (r) has finite order and ord(a) = ord (y(x)). Hint: to prove that ord(a) = ord ((r)), consider the two cases x = 1 and 1. For the second case, note that ord(x) > 1 and apply the second assertion of the Corollary to Proposition 1.3.3. Proof: (b) Prove that if x has infinite order, then (r) has infinite order. Hint: use the fourth assertion of the Corollary to Proposition 1.3.3.
4. Suppose (G₁, *) and (G₂, 0) are groups, let i and J denote the respective identities of G₁ and G2, let : G₁ G₂ be a group isomorphism, and suppose z E G₁. (a) Prove that if a has finite order, then (r) has finite order and ord(a) = ord (y(x)). Hint: to prove that ord(a) = ord ((r)), consider the two cases x = 1 and 1. For the second case, note that ord(x) > 1 and apply the second assertion of the Corollary to Proposition 1.3.3. Proof: (b) Prove that if x has infinite order, then (r) has infinite order. Hint: use the fourth assertion of the Corollary to Proposition 1.3.3.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.6: Homomorphisms
Problem 17E: 17. Find two groups and such that is a homomorphic image of but is not a homomorphic image of ....
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![4. Suppose (G₁, *) and (G₂, 0) are groups, let i and denote the respective identities of G₁ and G₂, let : G₁ → G₂ be a
group isomorphism, and suppose I E G₁.
(a) Prove that if z has finite order, then (r) has finite order and ord(z) = ord(v(z)).
Hint: to prove that ord(x) = ord(v(x)), consider the two cases x = i and z ‡ 1. For the second case, note that
ord(z) > 1 and apply the second assertion of the Corollary to Proposition 1.3.3.
Proof:
(b) Prove that if z has infinite order, then y(r) has infinite order.
Hint: use the fourth assertion of the Corollary to Proposition 1.3.3.
Proof:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18e7196c-e028-4487-9771-b214bf5d8f20%2F09bd561c-9257-40d1-ac84-863d630f641a%2F9p1eu4o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Suppose (G₁, *) and (G₂, 0) are groups, let i and denote the respective identities of G₁ and G₂, let : G₁ → G₂ be a
group isomorphism, and suppose I E G₁.
(a) Prove that if z has finite order, then (r) has finite order and ord(z) = ord(v(z)).
Hint: to prove that ord(x) = ord(v(x)), consider the two cases x = i and z ‡ 1. For the second case, note that
ord(z) > 1 and apply the second assertion of the Corollary to Proposition 1.3.3.
Proof:
(b) Prove that if z has infinite order, then y(r) has infinite order.
Hint: use the fourth assertion of the Corollary to Proposition 1.3.3.
Proof:
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