Many people believe believe sociopaths are proactively antisocial people. Part of the reason for that is that psychologists use the terms 'sociopathy' and 'anti-social personality disorder' synonymously. The terminology here can be confusing. The way I like to put it is this: A psychopath is a person who was born with weak emotions; a sociopath through conditioning lacks the ability to bond with others; and anti-social personality disorder is a clinical diagnosis for a person who lacks feelings for others and chronically breaks the law.
So if we're talking about sociopathy here, the condition has little to do with anti-social acts, nor does it have to do with narcissism. Sociopathy is a deep lack in the ability to connect with others emotionally, combined with a manipulative mindset. A person may or may not behave in an adversarial manner toward others as a consequence. Not connecting with others does not preclude you to treating them badly. The reason people often associate sociopathy with narcissism is because a person who does not connect with others tends to derive their self-worth not from their relationships with others, but from how they can compete with them. If they can compete well in the ways they think are important, they may become narcissistic.