**Otto Schreier** worked in combinatorial group theory, particularly on subgroups of free groups and on knot groups.

- He studied at the Vienna School of Technology and, from 1899 to 1906, worked as a partner with Ernst Lindner (1870-1956) in the architect firm 'Ernst Lindner and Theodor Schreier'.
- Otto studied at the Döblingen Gymnasium situated in the northwest of Vienna near the Vienna Woods.
- At the Gymnasium there were two students who were one year older than Schreier, namely Richard Kuhn (1900-1967) and Wolfgang Pauli.
- One year younger than Schreier was Karl Menger and the two became close friends.
- Schreier graduated from the Döblingen Gymnasium in July 1919 and, later that year, entered the University of Vienna to study mathematics.
- After a week of complete engrossment, he produced a definition of a curve and confided it to fellow student Otto Schreier, who could find no flaw but alerted Menger to recent commentary by Hausdorff and Bieberbach as to the problem's intractability, which Hahn hadn't mentioned.
- He kept in close contact with Schreier who sent his lecture notes to Menger.
- Life was becoming worrying for Schreier as the National Socialists began to cause trouble.
- Menger felt that his own independent contributions had not been acknowledged as much as they should have been so in 1926 he asked Schreier to verify the details of his contributions.
- This Schreier did in great detail giving dates and contents of all the correspondence he had had with Menger when he was in the sanatorium.
- Schreier's doctorate, supervised by Philipp Furtwängler, was awarded for a thesis Über die Erweiterung von Gruppen Ⓣ(On extensions of groups) on 8 November 1923.
- After receiving his doctorate, Schreier went to Hamburg and worked until his death at the Mathematische Seminar.
- It was Wilhelm Blaschke and Erich Hecke who recruited him to Hamburg having attended a lecture that Schreier gave at the 1923 meeting of the German Mathematical Society held in Marburg.
- However, even before this meeting, Schreier had been to Hamburg with Reidemeister and got to know the mathematicians there.
- Thus the young Schreier was already by then an all-round mathematician.
- Schreier gave lecture courses on group theory and analytic number theory, at the request of the mathematical faculty, before completing his habilitation.
- Hamburg was an exciting place for someone with Schreier's interests and, among others, he was influenced by Emil Artin.
- During the beginning of the 1928/29 session Schreier lectured on function theory giving parallel courses in Hamburg and Rostock.
- Although, as we have noted, many mathematicians influenced Schreier, the first were Furtwängler and Reidemeister.
- Schreier may have been directed towards the main theorem, which proves that certain torus knots were not isomorphic to their mirror images, by Reidemeister.
- These knots gave rise to groups which were free products with an amalgamated subgroup and Schreier studied this property in detail in a 1927 paper.
- However, Schreier will be best remembered for his work on subgroups of free groups which he studied in his habilitation thesis.
- In January 1926 Schreier attended a lecture given by Reidemeister in Hamburg on finding presentations for finite-index normal subgroups of finitely presented groups.
- Schreier, who was more interested in the algebraic implications of the method compared to Reidemeister's more geometrical interests, was able to extend Reidemeister's method to arbitrary subgroups and, by cleverly choosing generators for the subgroup, was able to greatly simplify the presentation obtained.
- These are now called Reidemeister-Schreier presentations and while using them Schreier was able to prove that subgroups of free groups are free.
- Schreier published his method in his 1927 paper Die Untergruppen der freien Gruppe Ⓣ(The subgroups of the free group).
- Schreier made important contributions to other parts of group theory.
- Schreier (1927) showed that the fundamental group of such a space is always abelian.
- Schreier (1928) found an important refinement of the fundamental Jordan-Hölder theorem, 39 years after the publication of Hölder's paper.
- The first plan was for Schreier and Artin to collaborate in publishing their lecture notes but Artin dropped out of the project.
- However, Schreier became ill and died before he could prepare his lecture notes for publication.
- Emanuel Sperner was a student at Hamburg who had been advised for his doctoral thesis by Wilhelm Blaschke but had been much influenced by Schreier's teaching and help.
- He stepped in to edit Schreier's lectures and to put them into book form.
- Let us end by noting that Schreier's death in 1929 meant that he died before Hitler's Nazi party came to power in 1933.
- Edith returned to Vienna where she was very close to Schreier's parents.

Born 3 March 1901, Vienna, Austria. Died 2 June 1929, Hamburg, Germany.

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Algebra, Group Theory, Origin Austria, Puzzles And Problems

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