Is the solvability of finite groups of order coprime to 15 essentially easier to prove than the entire Classification of Finite Simple Groups?
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As Derek Holt has pointed out, the answer to the question is yes.  Thompson proved that the only finite simple groups of order coprime to 3 are the Suzuki groups, and Glauberman later extended this to a classification of simple groups that do not have ${\rm S}_3$ as a subgroup. Both of these results are preclassification, though they might not have been published. 

