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One may conjecture that there exist only finitely many nonisomorphic noncyclic groups whose order g is divisible by exactly three distinct primes p<q<r. J. G. Thompson [ó] has shown that then p = 2, q = 3 while r is 5, 7, 13, or 17. I t is not unlikely that if one of the exponents a,b,c is 1, the methods applied here can be used to find all simple groups of the orders in question. No example is known in which all three exponents a,b,c are larger than 1. Since the proof of the theorem is long, we do not intend to publish it. A complete account has been prepared in mimeographed form. We shall give a brief outline.

On simple groups of order 5⋅3^{𝑎}⋅2^{𝑏}