Be warned, this sounds a little preachy but is not intended as such.
Anarchy, Greek origin means "without ruler" and an anarchist is someone who believes in Anarchism, a political philosophy which considers the state undesirable, unnecessary, and harmful, and instead promotes a stateless society, or anarchy. It branches of widely off from here as anarchism is a rather sweeping term.
So simply put an anarchist is someone who believes we shouldn't be governed buy the state or individuals but govern our own lives.
As with most things someone adapts imagery and ideals to their own ends and it becomes part of pop culture, in this case, punk rock of the 70's The A symbol is an A inside an O. Which stands for "Anarchy is Order" the punks adopted it as it was something of a rebellious symbol (typical punk "screw the system mentality") and this was intended as sensationalist marketing ploys, playing off of mainstream association of anarchy with chaos.
And that's where people get mixed up, they see anarchy as chaos and anarchists as people who want to kick ***, cause trouble or simply "screw the system" due mainly to a number of high-profile violent acts including riots, assassinations, and insurrections involving anarchists. Although most anarchist ideology condemns such acts.
If you want a punk cool, but call them a punk, not an anarchist, they are not the one and the same.
Phew, that was more than I wanted to write and I hope I've not made any mistakes.
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