Death penalty : Is this possible on XXI Century ?

A woman, Frances Elaine Newton was executed 18:09 h (23:09 GMT) by lethal injection at Huntsville, Texas . She was convicted in October 1988 and sentenced to die for killing her husband and two children on April 7, 1987. She always claimed to be innocent.(see USA Today).(Français )

Can a state (under the shield of the law) kill somebody? Isn't it also a crime ? Is this the example of a developed countrie? Pray and beg forgiveness for that people.