In the following question, out of the four alternatives, select the alternative which is the best substitute of the phrase.
A soldier or sailor, who rebels or refuses to obey the orders of a person in authority.

Let's look at the meanings of the given words:
mutineer: A person, especially a soldier or sailor, who rebels or refuses to obey the orders of a person in authority (Mutiny). This fits the definition perfectly.

radical: A person who advocates thorough or complete political or social reform.

anarchist: A person who believes in or tries to bring about anarchy (a state of disorder due to absence or non-recognition of authority).

revolutionary: A person who advocates or engages in political revolution.

Hence, mutineer is the correct one-word substitute.