He drove Trotsky out by a police force that he created in secret. Then, i think he killed Trotsky or sent him into exile.
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The main persons vying for power were Leon Trotsky and Joseph Stalin. There were others, such as Grigori Zinoviev and Nikolai Bukharin, but the first battle was between Trotsky and Stalin. Lenin favored Trotsky over Stalin, but Stalin was too entrenched in the Communist Party for Trotsky to win that struggle.
Trotsky never did reign over Russia. Lenin did at first after the Revolution. Then Lenin became ill and Joseph Stalin, Grigory Zinoviev and Lev Kamenev ruled together. After Lenin died, Trotsky was supposed to take over according to papers left behind by Lenin, but Stalin prevented that and forced Trotsky first out of the Communist Party then out of the country.
While Lenin was initially critical of both Trotsky and Stalin, toward the end he began to see Stalin's true ambitions and strongly favored Trotsky over him. Trotsky, (who was by no means an angle) was ruthless but extremely dedicated to communism and sincerely belived in the cause. Trotsky would probably no do any thing as awful as Stalin did and probably would have entered WWII before stalin. all and all Trotsky's soviet union would probably bear more resembelence to the one envisioned by comrade Lenin.
He was adverse to Stalin taking over, so he probably wanted Trotsky to come to power.
Stalin was glad to be rid of Lenin so that he could now take over the Communist Party and the Soviet Union himself. Stalin had been running the government behind Lenin's back with Grigory Zinoviev and Lev Kamenev. Now he did not have to hide it. Also, Stalin was making disparaging remarks about Lenin because of his invalided condition after his strokes. Trotsky was sad to see Lenin go, but at the same time worried because he knew what Stalin would try to eliminate him now that Lenin was no longer there to keep Stalin in line.