Facebook Poll – Anonymous Voting Capability

You don't want a group. You want a page.

http://www.facebook.com/home.php?#!/help/?faq=16140

Since groups look similar to Facebook Pages, what’s the difference between them?

Groups and Pages serve different purposes on Facebook. Groups are meant to foster group discussion around a particular topic area while Pages allow entities such as public figures and organizations to broadcast information to their fans. Only the authorized representative of the entity can run a Page.

It looks like this is currently somewhat broken.
If you go to an existing group (Aleph-One Strong for the Generalized Continuum Hypothesis!), then click on the category in the information, it takes you to a link which appears to be a search link (Outlandish Statements from the above group), but it gives you the message "Please enter a query." If you then put in a search term, it searches among all groups.

However, if you add &q=searchterm to the link, you get a search for your searchterm in your category. Taking the example from above, http://www.facebook.com/search.php?sfxp=1&c1=10&c2=141&type=groups&q=answer will get you a list of all groups in the category "Outlandish Statements".

With the millions of groups, and small number of categories for groups, a list of all groups in a category would be too large to look through them.