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 $\displaystyle G$ is finite. For $\displaystyle x\in G$ construct $\displaystyle \left < x \right>$. Then, $\displaystyle |x| = |\left< x \right>|\leq |G|$.
There are two solutions to $\displaystyle x^2 = 1$ in the multiplicative group but only one solution to $\displaystyle 2x=0$ in the additive one.Q2:Prove that (Q,+) is not isomorphic to (Q,*)?
If $\displaystyle G$ is infinite and $\displaystyle G = \left< a\right>$ then define $\displaystyle \phi : G\to \mathbb{Z}$ by $\displaystyle \phi(a^k) = k$ 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 $\displaystyle G$ is finite and $\displaystyle G = \left< a\right>$ with $\displaystyle |G|=n$ then define $\displaystyle \phi : G\to \mathbb{Z}_n$ by $\displaystyle \phi (a^k) = [k]_n$. Be sure to argue this is well-defined.