can a company hire you, have you work a week then fire you for a backround check?
Yes. But generally it is very inadvisable and sometimes it is actionable. Such as if the employer recruited you from another position or if the employer has used a discriminatory reason based on your membership in a protected class.
Lawsuit / Dispute Attorney
An employer can terminate you if there is a discrepancy on your application or if you credentials aren't what you claimed they were.
Gender, age, religious belief, race etc. all are discriminatory reasons for which you cannot be legally fired and upon which you might base a wrongful termination suit. If you are an at-will employee, you can be fired for any reason or no reason, except unlawful discrimination. If you do not have an employment contract or union to represent you, your recourse is limited. Discrimination based on age, gender, race, religious beliefs etc.... may give rise to a different answer.
You might find my Legal Guide helpful "Workplace Discrimination: A Basis for Wrongful Termination Claims"
You might find my Legal Guide helpful "How to Choose a Lawyer for you.”
You might find my Legal Guide helpful "What Do I Tell My Lawyer"?
No one can know what the record is in the case because online we cannot find out any details. If you have a discrimination theory about retaliatory discharge or hostile workplace conditions you'll need a lawyer. Check with a lawyer in your locale to discuss more of the details.
Good luck to you.
NOTE: This answer is made available by the out-of-state lawyer for educational purposes only. By using or participating in this site you understand that there is no attorney client privilege between you and the attorney responding. This site should not be used as a substitute for competent legal advice from a licensed professional attorney that practices in the subject practice discipline and with whom you have an attorney client relationship along with all the privileges that relationship provides. The law changes frequently and varies from jurisdiction to jurisdiction. The information and materials provided are general in nature, and may not apply to a specific factual or legal circumstance described in the question.