Peter Ludvig Meidell Sylow (/ˈsyːlɔv/) (12 December 1832 – 7 September 1918) was a Norwegian mathematician who proved foundational results in group theory. Sylow processed and further developed the innovative works of mathematicians Niels Henrik Abel and Évariste Galois in algebra. Sylow theorems and p-groups, known as Sylow subgroups, are fundamental in finite groups. By profession, Sylow was a teacher at the Fredrikshald Latin School (Norwegian: Fredrikshalds lærde og realskole) for 40 years from 1858 to 1898, and then a professor at the University of Oslo for 20 years from 1898 to 1918. Despite the isolation in Fredrikshald, Sylow was an active member of the mathematical world. 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. He was also elected into the Norwegian Academy of Science and Letters in 1868, a corresponding member of the Academy of Sciences in Göttingen and the University of Copenhagen awarded him an honorary doctorate in 1894.
Early life Ludvig Sylow was born in Kristiania (now Oslo) on 12 December 1832 to later minister and customs treasurer Thomas Edvard von Westen Sylow (1792–1875) and Magdalene Cecilie Cathrine Mejdell (1806–98). His father had been an officer and a captain in the cavalry, and later he served as the head of the Ministry of the Army between 1848 and 1854. Initially, his father was aware of his son's talent in mathematics, so he encouraged him to work independently. 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.
Career as a mathematician
Education and first steps in mathematics Sylow attended Christiania Cathedral School, graduating in 1850 after taking the examen artium. He then became a student at the University of Oslo where he began his studies of natural sciences. In 1853, the University of Oslo awarded him the Crown Prince's gold medal (Kronprinsens gullmedalje) for a Mathematics subject about Gnomonics. In 1856 he took the high school mathematics teacher's examination (Realkandidat, Norwish to Real candidate) with excellent grades. He completed his graduation in 1856, but since no university post was available, he taught for two years at Hartvig Nissen School, an independent girls' school in the Uranienborg district of Christiania, which had been founded by Hartvig Nissen and Ole Jacob Broch. His years there came during Broch's most energetic university period, and it was Broch who inspired Sylow to study Carl Gustav Jacob Jacobi's fundamental work on elliptic functions, among other things. In 1858, Sylow moved to the town of Fredrikshald (now called Halden) in Østfold county, where he taught at Fredrikshald Latin School (Norwegian: Fredrikshalds lærde og realskole) as the Head Teacher in Mathematics and Science, a modest position that he held for a whole 40 years, from 1858 to 1898, except for 1861 and 62 when he continued studies in Paris and Berlin and when he substitute taught for Broch at University of Christiania.
Abel and the theory of equations During his studies, Sylow had become interested in the work of Niels Henrik Abel, and especially in an unfinished work on equation theory that had been left behind. However, it was only at Hartvig Nissen School (1856–58) that he began to research that work more deeply, in part thanks to Ole Jacob Broch, who was the school's pure mathematics teacher at the time. It was Broch who 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, he managed to struggle through them and soon found that Abel had achieved a far deeper understanding of the theory of equations than what he had managed to write in his published papers. Some of Sylow's first attempts to publish some of Abel's unpublished results that he had found in his papers proved to be unsuccessful. For instance, he sent one of these papers to Crelle's Journal in Berlin, but the editor there, Leopold Kronecker, rejected the article. 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.
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