Illegals Pay Taxes?

Here's a question - if you're an illegal immigrant, how are you paying taxes? If you are illegal, you don't have a social security number. Perhaps I don't quite understand how taxes get paid, but the last time I checked you need a social security number for social security, income, and property taxes. Since illegals have no social security number, we are left with one of two situations:

1) They are not paying taxes, or

2) They are committing fraud.

Personally, I'm not sure that the illegal immigrant advocacy groups should be advertising the idea that massive numbers of illegal immigrants are committing fraud and identity theft by paying taxes.