To travel to another country, you will need a passport or refugee travel document. You may also need a visa to enter or exit a foreign country. An Officer will examine your green card and all other identification documents, including passports, I.D.s from foreign nationals, at a port-of-entry. You will be asked to present your green card and a US Driver's License.
A green card holder is allowed to travel anywhere in the world. Generally, temporary or brief trips outside the US do not affect your permanent resident status. You will lose your permanent resident status if you do not intend to move to the US as your permanent residence. It's whether you have been away from the United States for longer than one year.
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Although short trips to the US are generally not considered problematic, an immigration officer might consider whether your intent was to travel abroad for a temporary purpose. They may also consider whether you have maintained US family and community connections, maintained employment in the US, filed US income taxes, or established that your intention is to return to the US.
You may also need to provide evidence of temporary absence such as a US address, US bank accounts, a valid US driver's license, property in the US, or other documentation that can be used to support your claim. It is recommended that you apply for a Reentry Permit by filing Form I-131. If you are planning to be absent for more than one year from the US, it is best to do so before leaving.