problems on order??

**Mathventure** · September 7th 2009, 04:20 AM

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,+) ?

**ThePerfectHacker** · September 7th 2009, 07:53 AM

Originally Posted by Mathventure

Q1: let G be a group.For any x belongs to G,prove that |x|<=|G| ?

.

Assume $G$ is finite. For $x\in G$ construct $\left < x \right>$ . Then, $|x| = |\left< x \right>|\leq |G|$ .

Q2:Prove that (Q,+) is not isomorphic to (Q,*)?

There are two solutions to $x^2 = 1$ in the multiplicative group but only one solution to $2x=0$ in the additive one.

Q3:Prove that a cyclic group is isomorphic either to (Zn,+) for some positive integer n or to (Z,+) ?

If $G$ is infinite and $G = \left< a\right>$ then define $\phi : G\to \mathbb{Z}$ by $\phi(a^k) = k$ as the isomorphism. Be sure to argue this is well-defined.

If $G$ is finite and $G = \left< a\right>$ with $|G|=n$ then define $\phi : G\to \mathbb{Z}_n$ by $\phi (a^k) = [k]_n$ . Be sure to argue this is well-defined.

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