**Ludwig Sylow** was a high school teacher who proved what is perhaps the most profound result in the theory of finite groups.

- Thomas Edvard Sylow was a captain in the cavalry and later became a government minister.
- 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.
- Sylow attended Christiania Cathedral School, graduating in 1850.
- Sadly, although Sylow would have made an outstanding university lecturer, he did not make a particularly good school teacher.
- 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.
- 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- Sylow's first attempts to publish some of Abel's unpublished results that he had found in his papers proved a problem.
- Kronecker, therefore, rejected Sylow's paper but, of course, since Sylow had correctly portrayed Abel's unpublished results, eventually they were indeed attributed to Abel rather than Kronecker.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 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.
- 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 ppp has a subgroup of order ppp, Sylow asks whether it can be generalised to powers of ppp.
- 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.
- 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.
- 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 Ⓣ(Complete Works of Niels Henrik Abel).
- This funding allowed Sylow to take leave from his school for four years in order to devote himself to the project.
- Much additional Abel material was found and published in the Sylow/Lie edition which appeared on 9 December 1881.
- 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.
- Almost all work on finite groups uses Sylow's theorems.
- 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.
- In 1902 Sylow gave the welcoming address at a conference to mark the centenary of Niels Abel's birth.
- 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 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.
- Finally we should say a little about Sylow's life outside mathematics.

Born 12 December 1832, Christiania (now Oslo), Norway. Died 7 September 1918, Christiania (now Oslo), Norway.

View full biography at MacTutor

Algebra, Group Theory, Origin Norway

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive