[1] Sylow processed and further developed the innovative works of mathematicians Niels Henrik Abel and Évariste Galois in algebra.

He wrote a total of approximately 25 mathematical and biographical works, corresponded with many of the leading mathematicians of the time, and was an able co-editor of Acta Mathematica from the journal's start in 1882.

From home, Sylow learned a sense of duty and hard work, but was also taught to be modest and although this was done with the best of intentions, it would become an obstacle for him later in life since it meant that he was happy to spend many years in a more lowly position than he should have had.

[2] In 1853, the University of Oslo awarded him the Crown Prince's gold medal (Kronprinsens gullmedalje) for a Mathematics subject about Gnomonics.

[citation needed] In 1858, Sylow moved to the town of Fredrikshald (now called Halden) in Østfold county, where he taught at Frederiksborg Latin School as the Head Teacher in Mathematics and Science, a modest position that he held for a whole 40 years, from 1858 to 1898.

[1][2][3] Although Sylow would have made an outstanding university lecturer, he did not make a particularly good school teacher, since he was interested in the advanced areas of mathematics and had thus little enthusiasm for teaching at lower levels.

[1][2] Sylow showcased his discoveries at a Scandinavian meeting of naturalists in 1860 in Copenhagen, where he presented a solid interpretation of a strange equation-theoretic treatise by Abel, edited only in fragments.

[citation needed] In the following year, in 1862, Sylow lectured at the University of Christiania as a substitute for Professor Ole Jacob Broch, who had been elected to serve in the Storting, the Norwegian parliament.

[2] Among his listeners was the young Sophus Lie, who would later create a strange new science on the basis of these ideas, the theory of continuous symmetry.

[1][2] Lie once commented that Sylow deserved a university position because of his "broad knowledge, his sharp powers of criticism, and his outstanding mathematical work".

[2] When the famous French mathematician Camille Jordan published the standard work Théorie des Substitutions in 1870, Sylow was familiar with most of what was written there and more.

[1][2] Jordan was astonished and somewhat skeptical, but shortly afterwards, he wrote enthusiastically from Sweden, and he helped Sylow to get that 10-page thesis published that same year in 1872.

[2] Besides the thesis of 1872, Sylow's main work was the new edition of Abel's collected writings which he procured in association with former student Sophus Lie on a public basis, when in 1873, Sylow and Lie were commissioned to provide a new edition of Niels Henrik Abel's collected works, paid for by the state.

Further Abel documents had been discovered after the Sylow/Lie book came out in 1881 and, at the Third Scandinavian Congress of Mathematicians, which was held in Kristiania in 1913, Sylow discussed this new material.

[2] In 1883 Sylow became an editor of Acta Mathematica, was elected a member of the Academy of Sciences of Göttingen and, in 1894, the University of Copenhagen awarded him an honorary doctorate.

[1] As a result, Sylow spent a whole 40 years, from 1858 to 1898, holding the modest position of head teacher in mathematics and science at the Frederiksborg Latin School, a long reign that came to an end when Sylow was finally appointed as a professor of mathematics at the University of Oslo in 1898,[3] and despite already being 65 when he obtained a university post, he was still able to hold this position for 20 years, until 1918, when he died at the age of 85.