Q1: let G be a group.For any x belongs to G,prove that |x|<=|G| ?
Q2:Prove that (Q,+) is not isomorphic to (Q,*)?
Q3:Prove that a cyclic group is isomorphic either to (Zn,+) for some positive integer n or to (Z,+) ?
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Assume is finite. For construct . Then, .
There are two solutions to in the multiplicative group but only one solution to in the additive one.Q2:Prove that (Q,+) is not isomorphic to (Q,*)?
If is infinite and then define by as the isomorphism. Be sure to argue this is well-defined.Q3:Prove that a cyclic group is isomorphic either to (Zn,+) for some positive integer n or to (Z,+) ?
If is finite and with then define by . Be sure to argue this is well-defined.