Prometheos
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1. Prove that (Q,+) is not cyclic
Here is what I have, and I need help knowing if this proof makes sense, is thorough enough, or is completely wrong. Note that (Q,+) is rationals
Suppose, by contradiction, that (Q,+) is cyclic, p/q E (Q,+) and q=/=0
=> (Q,+) can be generated by <p/q> = {k(p/q)|k E Z}
if p/(2q) E (Q,+)
then p/(2q) can be generated by k(p/q)
=> p/(2q)=k(p/q)
1/2=k
therefore k does not belong to Z and (Q,+) is not cyclic
It seems like I am missing something or made a conceptual error somewhere. Any help or confirmation is appreciated.
Here is what I have, and I need help knowing if this proof makes sense, is thorough enough, or is completely wrong. Note that (Q,+) is rationals
Suppose, by contradiction, that (Q,+) is cyclic, p/q E (Q,+) and q=/=0
=> (Q,+) can be generated by <p/q> = {k(p/q)|k E Z}
if p/(2q) E (Q,+)
then p/(2q) can be generated by k(p/q)
=> p/(2q)=k(p/q)
1/2=k
therefore k does not belong to Z and (Q,+) is not cyclic
It seems like I am missing something or made a conceptual error somewhere. Any help or confirmation is appreciated.