How did the Russian mafia survive under Soviet Russia?

They survived and evolved in the prison system when Stalin was in charge. After he past away, all the prisoners who were locked up on Stalins order were released. The majority of the newly released prisoners did not become law abiding citizens but continued their mafia ways. They eventually began doing business with the Russian government officials which is when corruption within that communist party began to spike. The Russian mafia, with its infiltration within the communist government, played a huge part in which why the Soviet Government collapsed.

the KGB is the Russian mafia,they are the secret police

They didn't... they emerged after the fall of the Communist regime. You REALLY REALLY didn't want to be convicted of any crimes under the Soviets, they weren't all that concerned with "Innocent until proven guilty" "fair and speedy trials by juries of their peers" and all that... the only criminals who got away with organized crime during the Communist regime were the party officials who were corrupt and squandered millions of "communal money" for their own personal benefit.

They all worked at the KGB until the country collapsed.

The Russian Mafia didn't survive under Soviet Russia. They all came to America.

MONEY = POWER