Government is an institution with a monopoly of legitimized force in a given geographical region.
The mafia does not fit this definition. When a mafia don orders a person killed, people see it as illegitimate use of force. When an agent of the government kills a person trying to defend themselves from attack, it is seen as legitimate. The mafia does not fit this definition. When a mafia don orders a person killed, people see it as illegitimate use of force.