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Ludwig Sylow was a high school teacher who proved the most profound result in the theory of finite groups.
Sylow studied at Christiania University and won a mathematics contest in 1853. He then took the high school teacher examination in 1856 and taught in the town of Frederikshald from 1858 to 1898.
In 1861 Sylow obtained a scholarship to travel and visited Berlin and Paris. Lie had a special chair created for Sylow at Christiania University and Sylow taught at the university from 1898.
Between 1873 and 1881 Sylow and Lie prepared an edition of Abel's complete work. However Sylow's fame rests on one 10 page paper published in 1872.
In this paper Théorèmes sur les groupes de substitutions which Sylow published in Mathematische Annalen Volume 5 (pages 584 to 594) appear the three Sylow thoerems. Cauchy had already proved that a group whose order is divisible by a prime p has an element of order p. Sylow proved what is perhaps the most profound result in the theory of finite groups.
If p
is the largest power of the prime p to divide the order of a group G then
(i) G has subgroups of order p,
(ii) G has 1+kp such subgroups,
(iii) any two such subgroups are conjugate.
Almost all work on finite groups uses Sylow's theorems.
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