Proving G Abelian iff G/Z(G) is Cyclic

[DanielThrice]

This isn't some homework question its more theory based that I'm struggling with from class and we will probably have homework on it.

If G is some arbitrary group, why is G Abelian <---> the factor group of G/Z(G) is cyclic? My professor mentioned something about one direction being trivial but the rest of his proof was unfollowable.

 

[CompuChip]

Hi Daniel.

The trivial direction is the direct implication ("==>").
Suppose that G is abelian. Then every element commutes with every other element. So by definition then, Z(G) = G, and therefore G/Z(G) = {e}. The group with just one elements is definitely cyclic, so you are done.

Let me just get you started on the reverse, hopefully you can finish it by yourself. So suppose that G/Z(G) is cyclic. We want to prove that G is abelian. Since G/Z(G) is cyclic, this means that there is some a, such that G/Z(G) = {e, a, a², ..., aⁿ} (or possibly infinite). This means that every element from G can be written as a^k z for some value of k and some z in Z(G). Now can you show that gh = hg for any two such elements, g = a^kz and h = a^k' z'?

 

[DanielThrice]

Thank you compu chip i actually sat down and solved it, i thought the arrows mean't something else...been a late night. I got the other half of it, I can post it if you want it up for other users, unless you'd rather them try on their own

 

[CompuChip]

You can post it, if you would like us to check it.
I am confident that you managed to do it correctly though, and if you are too, you might as well save yourself the trouble of typing it out.

Just out of curiosity, what did you think the arrows meant?

 

[DanielThrice]

I thought it mean't implies that, I've had two professors use it diffeerently, which would be unidirectional...yea...one of those nights :)

 

[CompuChip]

Then they were sloppy and/or wrong.

"P <=> Q" always means "P implies Q and Q implies P", so if either is true then so is the other. Other formulations are "P if and only if Q", "P is necessary and sufficient for Q" and "P is equivalent to Q".