GND issue

I have a question on the ground connectors for you guys. On Arduino, there are 3 ground pins, one near the digital pins and two (side by side) near the analog pins. I supposed they're the same but maybe I was wrong. When I was powering my arduino witha 9V battery. The board did not work if I used one of the side-by-side GND pins but the LED directly on the 5V lit up. If I connected the battery to the other GND, both LED and the rest of the circuits (digital circuits driven by IO pins) worked fine. Did anyone have any strange experience with GND connectors at all? According to the diagram, all three are GND.

I supposed they're the same but maybe I was wrong.

No you are right, they are all electrically the same so you will have to look elsewhere to explain your observations. Can you repeat the effect?