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Many Cuban support their government because during the Batista Regime that governed Cuba was supported the American Government.  During this period, many people were persecuted and killed.  Apart from that, the Americans also controlled most of the businesses in Cuba.  Those who spoke out against Batista were silenced and many Cubans were now rallying against him.  He was overthrown in a revolution by Fidel Castro.  Many still remember the atrocities that were committed by Batista and they also resented the support the Americans gave to his administration.  Also Fidel Castro implemented many programs for the masses.  
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