You and her boyfriend did have an agreement. He paid the rent, and you let him live there.
The issue is what the agreement was. If the agreement was that he and his girlfriend would live there, then she was a party to the rental agreement, as well. That her boyfriend paid the rent does not affect the fact that she still owes it. If she is not paying the rent, you can give her a three-day notice. If she does pay, you can give her a 30-day notice.
On the other hand, if you had an agreement with the boyfriend alone and she just happened to live there without paying rent, she is a long-term house guest, and you could call the police. This scenario is very unlikely.
The answer to your question depends on the details and what you mean by "solely on the basis of no rental agreement." There are several laws which may cover the girlfriends status depending on the facts. You should seek the advice and services of a local landlord/tenant attorney if you wish the girlfriend to vacate the premises and she will not do it voluntarily.
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Yes, probably through the use of a 30 day notice.
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You cannot evict. Only a court can evict. You CAN terminate the oral agreement with proper written notice IAW your states' laws, if she refuses to leave, you will have to take her to court to evict.
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