Exercise 5.1.3. Decide whether each of the following claims are true or false. For each false claim, explicitly give the reason why it is false (for example, by listing which group axioms fail to hold in the given subset). (1) (Q*,·) < (R, +) {x € R* : |x| < 1} (3) (J,·)< (R*,·) where J = {x € R* : x² E Q} (2) (H,·) < (R*) where H =
Exercise 5.1.3. Decide whether each of the following claims are true or false. For each false claim, explicitly give the reason why it is false (for example, by listing which group axioms fail to hold in the given subset). (1) (Q*,·) < (R, +) {x € R* : |x| < 1} (3) (J,·)< (R*,·) where J = {x € R* : x² E Q} (2) (H,·) < (R*) where H =
Elements Of Modern Algebra
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![Exercise 5.1.3. Decide whether each of the following claims are true or false. For each false claim,
explicitly give the reason why it is false (for example, by listing which group axioms fail to hold in
the given subset).
(1) (Q*,·) <
(R,+)
(2) (H,·)< (R*) where H
= {x E RX : |x| < 1}
(3) (J,·)< (R\,·) where J = {x € R* : x² E Q}
(4) (K,o)< (Sn, 0) where K is the set of 2-cycles in S, together with the identity.
(5) (Z[i],+) < (C,+) where Z[i] = {a+bi : a,b E Z}.
(6) (3Z+1,+) < (Z,+). Here 3Z+1= {x €Z: x = 1 mod 3} = {..,–8, –5, –2,1,4,7,10,...}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c99a240-9de7-4947-9b54-53ff2cf6c85e%2F98a71761-872f-463e-968d-d44db0bdf945%2F6awk3w8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 5.1.3. Decide whether each of the following claims are true or false. For each false claim,
explicitly give the reason why it is false (for example, by listing which group axioms fail to hold in
the given subset).
(1) (Q*,·) <
(R,+)
(2) (H,·)< (R*) where H
= {x E RX : |x| < 1}
(3) (J,·)< (R\,·) where J = {x € R* : x² E Q}
(4) (K,o)< (Sn, 0) where K is the set of 2-cycles in S, together with the identity.
(5) (Z[i],+) < (C,+) where Z[i] = {a+bi : a,b E Z}.
(6) (3Z+1,+) < (Z,+). Here 3Z+1= {x €Z: x = 1 mod 3} = {..,–8, –5, –2,1,4,7,10,...}.
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