The American crime organization Molly Maguires

The American crime organization Molly Maguires
 

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The second-to-last one is correct because the comma is necessary.

The fourth sentence is correct.

My friend Howard and my wife Emma are cousins.
The use of commas in the above sentence it superfluous as it would be in the American crime organization the Molly Maguires.
Commas are used in parenthetical phrases in the following way: The Molly Maguires, an American crime organization, was active in the 19th century.

Read rule #4 here: http://grammar.ccc.commnet.edu/grammar/commas.htm

@poli: I agree that your first sentence should never contain a comma. I disagree that it and the sentence posted by Xinfu are alike. Xinfu's sentence contains an appositive, which is treated as a parenthetical phrase. Your sentence doesn't contain an appositive.

The other sentence you provided contains an appositive - 'an American crime organization'. You can switch the subject and the appositive around. Your sentence would then read "An American crime organization, the Molly Maguires, was active in the 19th century."

When the subject and appositive switched roles, the sentence still meant the same thing. The same logic can applied to the original sentence posted by Xinfu.
You can switch the direct object and its appositive around, like so:
"The story features the detective infiltrating the Molly Maguires, an American crime organization."

A parenthetical phrase can be removed and not change the essential meaning.
"The story features the detective infiltrating the Molly Maguires."
"The story features the detective infiltrating an American crime organization."

When the appositive is added, however, a comma must be used to offset it.

Thank you Rusty, Poli and Li~