# Finitely generated

• Oct 9th 2011, 10:35 AM
dwsmith
Finitely generated
A group H is called finitely generated if there is a finite set A s.t. $H=$

Prove that every finite group is finitely generated.

$=\bigcap_{\begin{cases}A\subset H\\H\leq G\end{cases}}H$

I don't know how to set that up with out the { and make it smaller under the intersection.

Anyways. Can I just say the intersection of finite elements is always finite? How can this be shown?
• Oct 9th 2011, 10:40 AM
Drexel28
Re: Finitely generated
Quote:

Originally Posted by dwsmith
A group H is called finitely generated if there is a finite set A s.t. $H=$

Prove that every finite group is finitely generated.

$=\bigcap_{\begin{cases}A\subset H\\H\leq G\end{cases}}H$

I don't know how to set that up with out the { and make it smaller under the intersection.

Anyways. Can I just say the intersection of finite elements is always finite? How can this be shown?

I'm not really sure what you're trying to get at. The idea is that a group is generated by a subset if (roughly) all elements in the group are just products of things in the set, and things whose inverse is in the set. What if you had a finite group $G$, what finite set has the property that every element of $G$ is a product of things in this set?
• Oct 9th 2011, 10:41 AM
Opalg
Re: Finitely generated
Quote:

Originally Posted by dwsmith
A group H is called finitely generated if there is a finite set A s.t. $H=$

Prove that every finite group is finitely generated.

$=\bigcap_{\left\{{A\subset H}\atop{H\leq G}\right.}H$

I don't know how to set that up with out the { and make it smaller under the intersection.

Anyways. Can I just say the intersection of finite elements is always finite? How can this be shown?

You can just take $A=G.$

Edit. Sorry Drexel28 (shouldn't that now be Maryland28?), I didn't see your reply.

Edit 2. Notice that I can make the { under the intersection smaller.