Unauthorized execution of order


Isn't it illegal for brokerage to do execution of unauthorized order? Recently I placed a buy order for 3000 stocks of PBLA at my brokerage which was executed in a minute. But within half an hour, I had an email saying that another buy order of 3000 PBLA stocks was executed. I clearly remember that I did not authorize the second order. I write down every order that I authorize. When contacted, they said they can do unauthorized sells if the client owes them money. And added that the transaction in question can't be undone!

Is there a way to prove that I did not authorize that order like any digital footprint that I leave or do not leave during placing the order from my home computer?

It's to note that my account is 'cash only', and I owe them nothing.
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