Should I forgive my cheating girlfriend?

Forgive her. Take her on her offer. Move on. You both love each other, its been 3 years. I was in a situation like this we still argue, you don't know what its like, I wish he understood too

Give it a little more time and it will get easier, one month is a still very little time after 2,5 years of relationship so it is normal that you miss her.
The reason why you should not take her back is that you two have completely different moral standards (or she doesn't have any I would even say) and this could never work in long run. She would keep on doing whatever she feels like (and probably behind your back) not considering your feelings and you would keep getting hurt.
I advise you to go on with your life, if you take her back then next time she might leave you and it will be so much worse for you.

I don't think you should unless you know she loves you and feels so stupid for doing it. If she did it more than once than I would not forgive or you could forgive and just forget too.

Nope. Don't take her back. An entire relationship with someone else shows you her character. She broke it off because it did not work out? Hello, calling all dip shits. Of course it didn't work out she was already in a relationship. How do you know she hasn't talked to him? Because she told you this? Well, she is really trusting. It is normal to not get her out of your head. You loved and trusted someone. It is heartbreaking I know. But, if you take her back the next time will be harder for you. It might be you that she breaks it off with and tells the other guy that it didn't work out. Selfish woman. I would just tell her "Hey way to step up and be an honest person, while I stayed faithful". Don't go back to her. Find a girl who you can be with that shows you and your relationship the respect you deserve. I was calling her a dip shit not you!

It really depends how you feel. If you really feel that you love her and life would be worse without her, then forgive her and give it another go.

However, you have to weigh up whether it is worth it to you emotionally, as her advice for you to cheat on her to "get even" is not morally correct and shows her character all too well.

I honestly can't tell you what to do, as this up to you. But one thing I can say, is don't cheat back, as I have been cheated on myself and it is not a feeling I would wish to inflict on someone else, especially if you love them.

Do you want to be with her? Because "love her" and "can't get her out of my head" doesn't necessarily mean you really want to continue now. But assuming you do (because you're considering it, because you asked the question. ) read on.

Don't take her up on the cheat offer. If you do, it will make it worse and more difficult. Now BOTH of you get to feel as crappy as you do today. And since you WANT to be with your girlfriend (see the assumption above), then why bother becoming a cheater? However if you seriously want to be with someone else, why stay with the girlfriend? Rather, if you want to cheat, instead simply end it with the girlfriend and have more fun with the next girl than you would cheating. Get it?

So, now, you want your girlfriend, and have chosen not to cheat. So now you're with your dishonest girlfriend who didn't feel guilty enough to break it off, and apparently would NEVER have been trustworthy if up to her. You'd better fix that, or that will eat you up forever. Since you can't change her, all you can do is tell her how you feel. If she cares, she'll be willing to try to change herself. If she doesn't care, well, I'd rethink your decision to want her (but stick with your decision to not cheat).

So now you want her, you don't cheat, and she really cares about how you feel and what she can do to make things better. Here's what you BOTH must do: talk it through to be totally clear on why she cheated, what was wrong, and what she needs from your relationship to never feel that way again. This will take a while. Many more conversations than one.

So now you want her, you don't cheat, she cares, and you both know specifically what was wrong and what to do to fix it. One more question, that you probably got answered during your multiple fun conversations about the problem(s) in your relationship: does she REALLY want you now? If not, well, go back to square one. If so, continue to the finish line.

So you BOTH want each other, you don't cheat, she cares, you BOTH know what was wrong, you BOTH want to fix it. Now go fix it. And the whole time, you have to be nice to her (even though she cheated) and never throw it in her face and mistreat her about it. You can feel insecure or angry, but you can't BECOME hurtful or vengeful because of your insecurities or anger. Ever. Or else (and especially with this particular girl) if you do, you'll just hurting her AND be creating a brand new whole in your relationship that she's likely to cheat to fill.

So there you go. The path to forgiveness. Is she worth it?