According the FWC:
Generally, alligators may be considered a nuisance when they are at least four feet in length and pose a threat to people or their pets or property. Alligators less than four feet in length are naturally fearful of people and are not generally capable of eating anything larger than a small turtle. They eat small fish, frogs, and other small animals. They are too small to be a threat to even small pets and pose no threat to people. They are typically not dangerous to people unless someone attempts to handle them. Also, they are common in Florida, and the mere presence of a small alligator is not cause for concern, even when they turn up in places where people may not expect to see them such as retention ponds and drainage ditches. However, occasionally alligators less than four feet in length are legitimate problems and must be addressed.
Some people would like the FWC to relocate nuisance alligators. While, in rare cases, the agency does allow for the relocation of alligators less than four feet in length, relocation of larger, problem alligators is not an effective solution. Relocation does not change the behavior that caused the alligator to be a nuisance, and relocated alligators frequently return to their capture site. As a result, these alligators would continue to pose a danger. Also, viable alligator populations exist in all suitable habitats in the state, making the relocation of alligators an unwarranted exercise from an ecological perspective. Relocation of wild animals can also facilitate the spread of disease.
Alligators commonly become a nuisance when they are fed by humans. Fed alligators lose their fear of people and can be dangerous. For this reason it is against the law in Florida to feed wild alligators or other wild crocodilians.
I too hate to see them destroyed but once a large one makes its way into areas frequented by humans, they really don't have much choice.