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  "body": "\n## Statement\n\nLet $p$ and $q$ be prime numbers, let $a, b \\geq 0$ be integers, and let $G$ be a finite [[group|G5S1M9]] of order $p^a q^b$.\nThen $G$ is [[solvable|0FXJ7YFMMX]].\n\n## Remarks\n\nA consequence is that the order of each finite [[simple group|0YMYMNQ0QZ]] that is not abelian has at least three distinct prime divisors.\nFor example, $A_5$ has order $60 = 2^2 \\cdot 3 \\cdot 5$.\nFor $b = 0$, the result says that finite $p$-groups are solvable, which follows from [[their nilpotency|081WTKYX6C]].\n\n## History\n\nWilliam Burnside proved the theorem in 1904, with the representation theory of finite groups.\nBurnside, Jordan and Frobenius had proved special cases before, in 1897, 1898 and 1902.\nJohn G. Thompson later showed that his work on the N-group theorem gives a proof without representation theory.\n\n## Explanations\n\n### L1 human\n\nIf the order of a group has only two prime factors, then the group is solvable.\nSo a simple group that is not abelian needs at least three primes in its order.\nThe smallest such group, $A_5$, uses the primes $2$, $3$ and $5$.\n",
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