Can a mother can give her child to adoption without the father permission?

First its not a good idea to piggyback your question on someone else's. This can lead to confusion. You should start a new thread. So I've moved your question to its own thread.

The answer is maybe. The courts will require that you get the father's permission. However, provision is made if the father can't be found. You will have to show a good faith effort to have gotten the father's permission before a court will proceed without it.

How old is the child?

If you know where the father is, normally yes, if the father wants the child, it is more likely father will get custody

And here's the thing:

If you LIE about not knowing the father and the father finds out--HE can overturn the adoption AFTER you have signed your rights away. So not only do YOU not get to have any say in the child (you have no parental rights), but the parents you chose don't get anything, and you'll end up paying child support for a child you can't have visitation to, custody of, or any say in the child's upbringing at all.

DO NOT LIE about the birthfather.