Given that ø is a isomorphism from a group G under addition to a group G under addition, convert property 2 of Theorem 6.2 to addi- tive notation. Suppose that o is an isomorphism from a group G onto a group G. Then 1. o carries the identity of G to the identity of G. 2. For every integer n and for every group element a in G, þ(a") = [ø(a)]". 3. For any elements a and b in G, a and b commute if and only if $(a) and 4(b) commute. 4. G = (a) if and only if G = (4(a)). 5. lal = l6(a)| for all a in G (isomorphisms preserve orders). 6. For a fixed integer k and a fixed group element b in G, the equation x* = b has the same number of solutions in G as does the equation x* = $(b) in G. 7. If G is finite, then G and G have exactly the same number of elements of every order.
Given that ø is a isomorphism from a group G under addition to a group G under addition, convert property 2 of Theorem 6.2 to addi- tive notation. Suppose that o is an isomorphism from a group G onto a group G. Then 1. o carries the identity of G to the identity of G. 2. For every integer n and for every group element a in G, þ(a") = [ø(a)]". 3. For any elements a and b in G, a and b commute if and only if $(a) and 4(b) commute. 4. G = (a) if and only if G = (4(a)). 5. lal = l6(a)| for all a in G (isomorphisms preserve orders). 6. For a fixed integer k and a fixed group element b in G, the equation x* = b has the same number of solutions in G as does the equation x* = $(b) in G. 7. If G is finite, then G and G have exactly the same number of elements of every order.
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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Transcribed Image Text:Given that ø is a isomorphism from a group G under addition to a
group G under addition, convert property 2 of Theorem 6.2 to addi-
tive notation.
![Suppose that o is an isomorphism from a group G onto a group G.
Then
1. o carries the identity of G to the identity of G.
2. For every integer n and for every group element a in G, þ(a") =
[ø(a)]".
3. For any elements a and b in G, a and b commute if and only if
$(a) and 4(b) commute.
4. G = (a) if and only if G = (4(a)).
5. lal = l6(a)| for all a in G (isomorphisms preserve orders).
6. For a fixed integer k and a fixed group element b in G, the
equation x* = b has the same number of solutions in G as does
the equation x* = $(b) in G.
7. If G is finite, then G and G have exactly the same number of
elements of every order.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25de9533-5920-415c-aa7a-f20a719c4f5e%2F6318abc4-8119-4478-b461-853e9d0fcd82%2Ft37h2y.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose that o is an isomorphism from a group G onto a group G.
Then
1. o carries the identity of G to the identity of G.
2. For every integer n and for every group element a in G, þ(a") =
[ø(a)]".
3. For any elements a and b in G, a and b commute if and only if
$(a) and 4(b) commute.
4. G = (a) if and only if G = (4(a)).
5. lal = l6(a)| for all a in G (isomorphisms preserve orders).
6. For a fixed integer k and a fixed group element b in G, the
equation x* = b has the same number of solutions in G as does
the equation x* = $(b) in G.
7. If G is finite, then G and G have exactly the same number of
elements of every order.
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