There are government, police, and military positions available to green card holders; however, some jobs are only available to United States citizens.
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Some police departments do require U.S. citizenship, other ones do not. Also, some branches of Government and military positions are open to LPRs, yet certain such positions like the FBI, BAFT, CIA and others do require the U.S. citizenship as an absolute pre requisite for employment eligibility with such federal agencies. You may inquire with each particular federal or state employer further.
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The attorney above attempted to provide a competent professional opinion, however, the law and its applications change frequently and vary greatly from other U.S. jurisdictions and locales.
Therefore, any information and opinions expressed are general in nature, and may not apply to specific, factual or legal circumstances related to one's present legal issues.
Contact an experienced lawyer admitted to practice in that State under an attorney-client privilege to further receive comprehensive legal assistance before making an educated decision about your particular legal issue.
Respectfully, Attorney Alexander Ivakhnenko, Chicago, Illinois