Although Sylow had a good upbringing, learning how to work on his own, being taught the importance of doing one's best and working hard, nevertheless in some ways his upbringing would prove a disadvantage in his career. This was because he was taught to be modest and although this was done with the best of intentions, still it meant that he was happy to spend many years in a more lowly position than he should have had. It is interesting to note that in 1902, when Sylow gave the welcoming address at a conference to mark the centenary of Niels Abel's birth, he said (see ):-
... such a great modesty maybe does not fit this world, it may also be seen as a weakness.Sylow attended Christiania Cathedral School, graduating in 1850. He then began his studies of natural sciences at Christiania University where won a mathematics contest in 1853. He then took the high school teacher's examination in 1856, qualifying as a teacher of mathematics and natural science with excellent grades. As no university post was available, he taught for two years at the at Hartvig Nissen school. He then moved to the town of Fredrikshald (now called Halden) in Ostfold county, where he taught at secondary school from 1858 to 1898. Sadly, although Sylow would have made an outstanding university lecturer, he did not make a particularly good school teacher. He was interested in the advanced areas of mathematics and had little enthusiasm for teaching at lower levels. Also he found it difficult to keep discipline in his classroom so the fact that his career was largely in schools rather than universities was a poor use of his talents on two scores - universities were the poorer for not having Sylow as a lecturer, while schools were poorer for having him as a teacher!
Sylow continued his mathematical studies however (see ):-
... at first working on elliptic functions in the tradition of Abel and Jacobi, inspired by the professor of pure mathematics Ole Jacob Broch. Finding Abel's papers on the solvability of algebraic equations by radicals more interesting, Sylow was led from them (by the professor in applied mathematics, Carl Bjerknes) to Galois.Ole Jacob Broch had studied mathematics at Christiania University after which he travelled to Paris, Berlin and Königsberg where he studied a range of mathematical topics, particularly optics and statistics. After returning to Norway, together with his friend Hartvig Nissen (1815-1874), he founded the Hartvig Nissen school, an independent girls' school in the Uranienborg district of Christiania. Broch became a lecturer in applied mathematics at Christiania University in 1848, but took over Bernt Holmboe's pure mathematics teaching in 1850 following his death. Broch retained his interest in the Hartvig Nissen school, of which he had been a co-founder, and gave the young teacher Sylow much encouragement to continue his advanced mathematical researches. Although at first Sylow found reading Abel's papers a difficult task, soon he found that Abel had achieved a far deeper understanding of the theory of equations than his published papers indicated. Sylow's first attempts to publish some of Abel's unpublished results that he had found in his papers proved a problem. This was due to the fact that he sent a paper containing these results to Crelle's Journal where the paper was considered for publication by Leopold Kronecker. Now the difficulty arose since Kronecker had already published these results having discovered them himself, and had no wish to have a paper in print which showed that Abel had proved them long before he had. Kronecker, therefore, rejected Sylow's paper but, of course, since Sylow had correctly portrayed Abel's unpublished results, eventually they were indeed attributed to Abel rather than Kronecker.
In 1861 Sylow obtained a scholarship to travel and visited Berlin and Paris. In Paris he attended lectures by Michel Chasles on the theory of conics, by Joseph Liouville on rational mechanics and by Jean-Marie Duhamel on the theory of limits. He says, in the report he wrote at the end of the scholarship, that he also:-
... made myself acquainted with newer works, particularly in the theory of equations.In Berlin he had useful discussions with Kronecker but was unable to attend courses by Karl Weierstrass who was ill at the time. Bent Birkeland writes in  that, as there were no courses being given in Berlin that interested him:-
... he worked instead in the library, studying number theory and the theory of equations. He also got acquainted with Carl Borchardt, the editor of 'Crelle's Journal' ... It is interesting to note that no lectures in algebra or the theory of equations are mentioned from his stay either in Paris or in Berlin.In 1862 Sylow lectured at the University of Christiania, substituting for Broch who had been elected to serve in the Storting, the Norwegian parliament. In his lectures Sylow explained Abel's and Galois's work on algebraic equations. A summary of these lectures is presented in . It is worth noting that although he had not proved 'Sylow's theorems' at this time (he published them 10 years later) he did pose a question about them. After proving Cauchy's theorem that a group of order divisible by a prime p has a subgroup of order p, Sylow asks whether it can be generalised to powers of p. These lectures are significant for several reasons, not least that a young student attending them was Sophus Lie. Sylow's lectures were extremely valuable in giving Lie a fundamental appreciation of a topic to which he would make major contributions. Broch was again in the Storting from 1865 to 1868 and he was keen to have Sylow take over his university teaching during this time. However, the school in Fredrikshald refused to give Sylow leave to teach at the university. In 1869 Broch left his chair of pure mathematics, leaving a vacancy that Sylow was well qualified to have filled. However, the University of Christiania did not rate pure mathematics very highly at that time, preferring more practical and applicable topics. Sylow was too theoretical in his approach so he was not appointed.
In 1870-71 Sylow exchanged nine letters with Julius Petersen who, at this time, was working on his doctoral dissertation. Petersen sought Sylow's advice about the main theorem of his dissertation and these letters all deal with this. The article  discusses these letters. The two mathematicians exchanged another sixteen letters in 1876-77. Between 1873 and 1881 Sylow and Lie prepared an edition of Abel's complete work published under the title Oeuvres complète de Niels Henrik Abel . The motivation for this had come from the Norwegian Academy of Science who applied to the Norwegian Parliament for funding for the project which was quickly granted. This funding allowed Sylow to take leave from his school for four years in order to devote himself to the project. Arild Stubhaug writes in :-
In the course of the work Sylow had an eager discussant in C A Bjerknes, who for his part, worked on the Abel biography that was published in 1880. Bjerknes wanted as much as possible of Abel's early works to come out, not only his great treatises with their exemplary stringency, and perhaps there had been more Abel material than what Holmboe used in his edition of 1839?Much additional Abel material was found and published in the Sylow/Lie edition which appeared on 9 December 1881. Lie said that most of the work was done by Sylow and commented that Sylow deserved a university position because of his:-
... broad knowledge, his sharp powers of criticism, and his outstanding mathematical work.However, today 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 theorems although we know that he had already proved his famous theorem by September 1870. Cauchy had already proved that a group whose order is divisible by a prime p has an element of order p. Sylow generalised this, proving what is perhaps the most profound result in the theory of finite groups:
- G has subgroups of order pn
- G has 1 + kp such subgroups
- any two such subgroups are conjugate.
Sylow became an editor of Acta Mathematica, was elected a member of the Academy of Sciences of Göttingen in 1883, and, in 1894, was awarded an honorary doctorate from the University of Copenhagen.
Lie had a special chair created for Sylow at Christiania University and Sylow taught at the university from 1898. Note that he was 65 years old before he obtained a university post but, remarkably, he was still able to hold this position for 20 years. G A Miller writes :-
Notwithstanding the advanced age at which Sylow entered the university faculty he is said to have filled the position during twenty years with marked success. The duties of his professorship did not seem to be burdensome to him until the last year of his life when he frequently remarked that he felt tired.Although at the age of 65 he had at last become a university professor, we should note that he did not receive the salary of a professor. At first, he was paid a headmaster's salary which was approximately half the salary of a university professor. Later he received salary increases.
In 1902 Sylow gave the welcoming address at a conference to mark the centenary of Niels Abel's birth. He said (see ):-
In the early nineteenth century, applied mathematics had already achieved great triumphs, especially in the fields of astronomy and physics. But just at the same time mathematics ... started to turn its gaze back to the pure and abstract theories. [Gauss and Cauchy] initiated that great movement, which has run through the whole of the previous [19th] century, and which has reformed mathematics from its foundations at the same time it enriched it with new theories. ... It was in this movement that Niels Abel took such a significant part that he will forever he counted as one of the greatest mathematicians ever.In 1902 Sylow, in collaboration with Elling Holst, published Abel's correspondence. 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.
In  G A Miller quotes Georg Frobenius's opinion of Sylow:-
[Sylow] was known to mathematicians of every civilised country on account of a well-known theorem that bears his name. In 1876 Frobenius remarked that "as every educated person knows the Pythagorean theorem so does every mathematician speak of Abel's theorem and Sylow's theorem".We must not give the impression that the Sylow theorems and the Abel material were Sylow's only mathematical contributions. He also published a few papers on elliptic functions, particularly on complex multiplication, as well as papers on group theory. Finally we should say a little about Sylow's life outside mathematics. He never married but was a warm person with a nice sense of humour. He was an avid lover of being out of doors and often spent summer vacations in the mountains, usually in Kongsvoll, where he studied plants. Kongsvoll is a mountain station providing food and shelter on the route between Oslo and Trondheim, erected when the route was used by pilgrims visiting the shrine of St Olav in Trondheim.
Article by: J J O'Connor and E F Robertson