# Thread: Is this group finitely generated? (Omega_infinity)

1. ## Is this group finitely generated? (Omega_infinity)

if $\Omega_n:=\{ z\in \mathbb{C} : z^n=1 \}
$
, then it's a cyclic group with an order n, and one generator $(cis\frac{2\pi}{n})$.

Does $\Omega_\infty$ have a finite number of generators? (How do you prove \ disproof this?)

if $\Omega_n:=\{ z\in \mathbb{C} : z^n=1 \}
$
, then it's a cyclic group with an order n, and one generator $(cis\frac{2\pi}{n})$.

Does $\Omega_\infty$ have a finite number of generators? (How do you prove \ disproof this?)
If you mean the set of elements of infinite order, then the answer is,

No, $\Omega_{\infty} \equiv \mathbb{R}$. I cannot conjour up a `neat' proof, but I can direct you here, wikipedia's article on the Circle Group, the group of complex numbers with absolue value 1. This group is isomorphic to $\mathbb{C}^\times$.

3. Hmm.. I can understand that every element in Omega-Infinity 'looks like' a+bi, when both a,b are real (and cover all the real numbers). I think it's enough to say that it's order is bigger or equal to the order of R, which is not finitely generated, isn't it?

Hmm.. I can understand that every element in Omega-Infinity 'looks like' a+bi, when both a,b are real (and cover all the real numbers). I think it's enough to say that it's order is bigger or equal to the order of R, which is not finitely generated, isn't it?
Yes, of course! That is a very neat proof, but also quite subtle.

What you need to prove is that $\Omega_{\infty}$ is uncountably infinite, that it has cardinality greater than the natural numbers. This is sufficient because a finitely-generated group will always be countable (you should prove this - everyone who studies group theory should prove this! This bit is the subtle bit).

5. It really reminds me of set-theory, with Aleph-0, and Aleph (as orders of sets - naturals, integer, etc...)

I will try solving this, it looks a little 'heavy' right now, and the question this post relies on isn't 'too demanding', it only asks whether this group is finitely generated or not.

Thank you very much