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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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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