We had classes together a year ago, we never talked then. Now I have her again. She recognized me. I talked to her once, for a while. We aren't friends. I saw her facebook, she is single. How do I get her number? How do I show her alpha male dominance, mental dominance? The problem is I only see her once a week, (thats when my class meets), and we work in separate groups. Any ideas?

You talk to her.

As for the other questions you need to start reading some PUA logs or books, or go about experimenting.

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Most pua stuff applies. I pull a few numbers in class each semestr and I would give you some pointers.

1)Don't gravitate towards her
It's easy to want to sit next to them each class. If you end up near each other by chance or she sits near you that's fine. Even sitting near them is fine as long as you got a game plan/routines and can avoid smothering her.

2)socialize with everyone
Try and find commonalities with other classmates and have interesting convo (storytelling with DHV). Talking to the professor (not just asking questions) is a great way to convey your personality, since everyone will have their eyes/ears on you two. You want to be at the point where you're engaging a large set causing everyone in proximity to eavesdrop (including the target) out of curiosity, giving you social proof.

3) go deeper
The most common convo started in college is the "What's your major?" Thread. Most people ask this off the bat, but its probably best for rapport/comfort. The plus you can lead into the emotions driving her towards her major to build comfort, instead of "how many units you got
left? Did you take chemistry? Blah blah blah"

4) don't rush
Its a foregone conclusion you'll see this girl again, so don't rush into #closing. I've pulled a couple numbers at the A3 phase and they always ended up as flakes, despite the fact they were attracted. This leads to awkward moments in the next class which can work against you if it happens with too many girls. Make sure you have some deeper rapport than knowing her major which anyone could find out in 30 seconds.

These are the obvious ones that come to mind.

I don't know, I'm in a similar situation kinda. First impressions are everything and you spent the last year not speaking to her. It's kind of hard to break that.

Go and ask for a phone number. It is not hard to do

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Dubli

No need to be dominant. Don't know where you got that business from.

You have to talk to her. Small talk, discuss class and school, maybe even current events. Build a little rapport. Find out a little more about her. Ask her if she is on facebook, then friend her there.

Get her number in a casual way. Make it seem like you want to call her about an assignment or something. Then go from there.

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“Nothing is impossible, the word itself says 'I'm possible'!” ~Audrey Hepburn

you want dominance? invite people to come out with you in your class. being a boss gives you value.

class openers i usually open with.
how did you're -blank- go? then make a statement that relates why your interested in it.
how is your work group it looks like you have a lot of bologna heads over there.

facebook is viable if she has under 200 friends. If she has over 400+ don't join her fan base.

asking for her number i like this: hey give me your number, we can do -blank-.

RedCali has some pretty good ideas as well I agree with him.