Ignoring how it would be absolutely impossible, Trotsky may or may not pull the Great Purge off.
I'd say it was pretty unlikely, as the man wasn't as paranoid as Stalin and extremely pragmatic in his approach for former Tsarist officers, but he may be forced into it by a declining economy and ailing foreign policy.
Economically, the man couldn't do Stalin's Five Year Plans, they required directing resources that could go to spreading socialism overseas into socialism in one country, which he hated. He also hated Lenin's NEP, which probably puts him into incompetent economics territory. Adding to that, he opposed any leaving of the gold standard (which is a terrible idea, fiat currency is far better for economic management)
Foreign policy wise, the man would probably try forcing a socialist revolution onto countries with no hope of actually pulling it off, and probably leading to international isolation. He absolutely would not make deals with the Germans or invade other countries - he wrote a lot of anti-Stalin polemics about that.
