The short answer to 'what is Trotskyism?' would be 'revolutionary democratic socialism'. Trotskyists struggle for socialist/communist revolution while putting emphasis to workers power and democracy, and aim for a society with a democratically planned economy, as opposed to a bureacratically planned one like the Stalinists or a market economy like capitalists.
The ideology is named after Leon Davidovich Trotsky, one of the leaders of the Russian revolution in 1917, the founder and first leader of the Red Army and one of the leading figures of the early Soviet state.
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So what is the difference between Trotskysm and Leninism.. Many people combine Marxism and Leninism.. can you combine Marxism-Trotskysm ?
Rosario is correct. Trotsky considered himself an orthodox Marxist and a Leninist, so in fact Marxist-Leninist would describe his views. The word is however traditionally associated with the Stalinists, which is why Trotsky used Bolshevik-Leninist and Trotskyists today usually call themselves either Marxists or Trotskyists.
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Frankly, I don't know what Marxism is. I get what communism is, but I'm not informed about the specifics.
Karl Marx was the author of Das Kapital and the Communist Manifesto (together with Friedrich Engels), which are perhaps the most important sources of socialist/communist thought. He was also one of the founders of the first proper communist (marxist) organisation, the Communist League. Thusly he is considered the 'father' of modern (non-anarchist) socialism/communism.
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I'm guessing there is difference becouse of the vanguard (I also don't know what that is)
You aren't completely wrong there, although the difference is between Trotskyists and Stalinists ('marxist-leninists'), not 'Marxists'. But it indeed has to do with the vanguard, or more specifically it's role. The vanguard is the ideologically most advanced, revolutionary layer of the working class. Marxists aim at organising this vanguard into a revolutionary party.
Now, as said Trotskyists put emphasis on democracy and believe that this party should be accountable before the workers, and it's representatives electable and recallable and without privileges that ordinary workers don't receive. As can be seen of the development of the USSR and other Stalinist countries, Stalinists don't share this idea. Instead an unaccountable and undemocratic bureacracy has invariably seized power in Stalinist parties and the countries they controlled and still control.
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and should socialism be in one country...
Socialism in one country was Stalin's main theoretical contribution to socialist thought, one that we trotskyists argue was a huge mistake. Trotsky and the Left Opposition wanted the USSR CP (also known as the Bolshevik Party) to continue the Marxist tradition of internationalism, and actively work to spread the revolution to other countries -- most importantly Germany and the rest of Europe -- as Russia was too technologically backwards to stand alone against the capitalist world or to develop socialism.
Had this been done, we would not only have avoided avoided the horrors of fascism and the second world war, but the world would most likely be socialist in it's entirety by now. Instead, precisely as Trotsky predicted would happen, the Stalinist bureacracy finally reinstalled capitalism in the USSR.
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And why did Stalin dislike Trotsky ?
Trotsky opposed the bureaucratic development in the USSR, and formed together with his supporters a fraction known as the Left Opposition within the Communist Party of the USSR. Stalin, however, managed to skillfully outmanouver it's members from the party's Central Committee (as he later did with anyone opposed to his rule) and later from the party itself.
Trotsky was expelled from the country and went to exile, ending up in Mexico where he was murdered by Stalin's agents while he was organising an international resistance movement against Stalinism and for a democratic socialist world revolution, known as the Fourth International.
It has since split into several 'competing' Internationals which link up Trotskyist parties all around the world. The largest one is called United Secretariat of the Fourth International (USFI), the second largest Committee for a Workers' International (CWI). Other notable Internationals are International Marxist Tendency (IMT) and International Socialist Tendency (IST).
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This thread should be in the Learning section if OI;
moved.