there was no socialism in east-germany. there was none in east-europe. that was fascism with a tiny bit of communism-appearence thrown in. socialism is found in scandinavia, belgium, netherlands, france, and the former western-germany.
Most Western European countries are mixed economies, mostly capitalist, with some socialism, and a welfare state.
East Germany and the Soviet Union really bought into the idea of Socialism: the state owned everything. Private property was outlawed. You could go to jail for making a profit.
The East Germans were so committed to the idea that the state owned everything that they believed they had a right to build an enormous wall to keep the governments property (people) from escaping to the West.