She didn't forget who you were, that was just an excuse to be a bitch because she wanted to sleep with someone else. Even if alcohol was an excuse, if you can't handle yourself while drinking, don't drink. Either way she's a terrible person, cut your losses OP, you don't want to be with a person like her.

How long have you been dating? Doesn't seem like shes not taking you or your relationship seriously. FYL, sorry mate.

If she's that drunk that her memory was impaired, I'd like to point out that legally, she probably couldn't consent to going home with that guy, let alone to any activities they very likely participated in. Which, if the guy heard her say she didn't know who you were, but knew you were her boyfriend, means that he would have been taking her home fully aware that she was far too under the influence to consent. The nature of this FML hinges on whether OP's GF really was so drunk that her memory recall failed in regards to OP- if she was, then the guy that took her home knowing that is the bigger problem, especially if he was encouraging OP's GF to drink. If she wasn't that drunk, both the GF and the guy are at fault in equal proportions. Though, OP is under no requirement to stay with the GF, regardless of the circumstances of the event. I just feel the need to point out that there are more issues to this than if the person cheats while perfectly sober or only mildly influenced by any kind of drug.

Time for a new girlfriend. If she does that in front of your face, imagine when she has a "girls night out" where you're not invited. Please move on now and save yourself from emotional damage trying to "control her" into being faithful. A girlfriend should automatically provide positive value to your life at minimum without you having to compel her to do so. ask yourself: does her behavior comport with this ideology? If not, you know what to do.

I am very sorry OP. Girls like her aren't worth your time or effort though