Consequences of Polygamy

The practice of polygamy usually does not go to trial, but there have been charges of bigamy. Polygamists are usually charged with statuatory rape if anything.
An article in the New York Times states "Usually the Debate, begun in the newspapers, languidly taken up in the pulpits, and finally dying away in the halls of congress, has not seriously disturbed the mormon magnates."
I did not find the exact situation Tom had questions about, but I found something interesting. A man of the United States, Pat, marries Lina, an immigrant. Lina gets her green card, as you would guess. Then, Pat also marries Bethany, another immigrant. Lina will not lose her green card in this event. However, if Bethany tries to get her green card, her green card will be denied based on polygamy. This is just a hypothetical situation obviously, but I figured it would be an easier way to explain the law.