**Otto Schreier**'s parents were Theodor Schreier (1873-1943) and Anna Turnau (1878-1942). Theodor Schreier was born in Vienna, the son of the businessman Moritz Schreier and his wife Regina. Theodor was one of a large family having siblings Berthold, Rudolf, Alois, Max and Marie. 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'. After 1906 he set up his own architect business and undertook major projects for the public sector such as schools, synagogues and office buildings. His most important work was designing the synagogue in St Pölten which was built in 1912-13. Theodor married Anna and they had one child, Otto, the subject of this biography. The family was Jewish.

In 1914, when Otto was thirteen years old, World War I broke out. Otto's father, Theodor, had undertaken military service before beginning to work as a partner in the architect's firm and, at the outbreak of war, he served in the Militärbaukommando which was involved in military construction. Otto studied at the Döblingen Gymnasium situated in the northwest of Vienna near the Vienna Woods. This school, which was excellent for mathematics and science, took a new approach emphasising English rather than Latin, Greek and French. This was not an easy time to be growing up in Vienna for, after the outbreak of World War I, in September 1914 the Döblinger Gymnasium building was converted into a war hospital, and teaching was transferred to a secondary school building in the Krottenbachstrasse. In February 1916 the war hospital closed and pupils returned to the original building.

At the Gymnasium there were two students who were one year older than Schreier, namely Richard Kuhn (1900-1967) and Wolfgang Pauli. Both Kuhn and Pauli went on to win Nobel prizes; Kuhn was awarded the Nobel Prize for Chemistry in 1938 and Pauli was awarded the Nobel Prize for Physics in 1945. 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. At Vienna he attended lecture courses by Wilhelm Wirtinger (on function theory and differential geometry), Philipp Furtwängler (on algebra and number theory), Hans Hahn (on set theory, and on real function theory and set theoretic geometry), Kurt Reidemeister (on combinatorial topology), Tonio Rella (1888-1945), Josef Lense (1890-1985) and Leopold Vietoris. Schreier became even closer to Menger and he was the person that Menger turned to when he believed he had a major contribution to the notion of dimension [11]:-

In the first lecture Hahn formulated the problem of making precise the idea of a curve, which no one had been able to articulate, mentioning the unsuccessful attempts of Cantor, Jordan, and Peano. The topology used in the lecture was new to Menger, but he "was completely enthralled and left the lecture room in a daze". 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.

Schreier told Menger that Hausdorff had written, "the sets traditionally called curves are so heterogeneous that they do not fall under any reasonable collective concept," while Bieberbach had written, "anyone trying to define the concept of a curve certainly would need a description as long as a tape worm, and of Gordian entanglement." Menger approached Hahn who [11]:-

... after some thought agreed that Menger's was a promising attack on the problem.

However, soon after this Menger became ill and went to a sanatorium in Aflenz in the mountains of Styria in southern Austria. He kept in close contact with Schreier who sent his lecture notes to Menger. Schreier wrote about the lectures he was attending:-

Hahn's lecture is extremely beautiful, although of course not such a polished product as his lectures from last winter. ... Furtwängler did not get very far in his seminar, now he is treating the inessential discriminant divisors ... Very pretty paper on Fourier series in Hahn's seminar...

Life was becoming worrying for Schreier as the National Socialists began to cause trouble. For example Schreier wrote to Menger on 28 November 1922 (see [2]):-

Today I received your kind letter with the new proof of the invariance theorem, which seems to me not only completely correct, but also much more transparent than the previous one ... . I will deliver the letter and the paper to Hahn tomorrow, provided the German Nationalist students permit it. Indeed, all University institutes, including ours, have been occupied by certain student groups, who also obstructed the lectures.

Menger's work on dimension was quite independent of that by Pavel Samuilovich Urysohn who sadly drowned in 1924 before publishing his results. 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 had also told Menger in the same letter of November 1922 about the new young lecturer who had just been appointed to Vienna, Kurt Reidemeister:-

The new geometer Prof Dr Kurt Reidemeister is an accomplished person. He is still very young(at most28, I would guess), full of wit and high spirits. He has been recommended by Blaschke. In addition to the elementary course on analytic geometry he lectures two hours per week on topology. I attend this lecture, of course, although its confusion exceeds even Wirtinger's worst ... Reidemeister's lecture at the Mathematics Society was very pretty, although one absolutely could not follow towards the end. By his humorous remarks he caused such roaring laughter as has never been heard, so it seems, in the Mathematics Society.

Schreier's doctorate, supervised by Philipp Furtwängler, was awarded for a thesis *Über die Erweiterung von Gruppen* Ⓣ on 8 November 1923. In this work he tackled a very fundamental group theory problem that had been posed by Otto Hölder:

Given two groups

GandH, find all groupsEhaving a normal subgroupNisomorphic toGsuch that the factor groupE/Nis isomorphic toH.

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. At Hamburg he was appointed to the post of assistant in the summer 1925 and worked for his habilitation. He was happy in this environment, writing to Menger shortly after taking up the position (see [2]):-

I even was invited by Blaschke for dinner on Sunday evening; things were very festive. Reidemeister entertained the whole society, and there was some playing of music... As soon as I can find time I will visit the harbour, which must surely be very impressive.

Although he was pleased to be appointed as an assistant, he received no salary for at least the first year there and he could only live thanks to financial support from his parents. He shared a flat in Hamburg with Heinrich Behnke who had been appointed as an assistant at Hamburg in 1922. Behnke later wrote about his friend (see [2]):-

He brought with him Viennese culture in the best sense. He was an enthusiastic disciple of Hans Hahn, but had obtained his doctorate with Furtwängler through a work on group theory. In addition, his ideas had been greatly stimulated by Wirtinger, the most renowned Austrian mathematician at the time. Thus the young Schreier was already by then an all-round mathematician. In addition, he was as gifted in music as in mathematics.

In fact Schreier gave lecture courses on group theory and analytic number theory, at the request of the mathematical faculty, before completing his habilitation. This was formally awarded on 1 December 1926 for a thesis entitled *Die Untergruppen der freien Gruppe* Ⓣ. For his probationary lecture, a necessary part of habilitating, he chose to give *On the concept of a curve*. His friendship with Menger clearly influenced his choice of topic which had to be understandable by non-mathematicians. However he had been appointed to a salaried position earlier, in fact in April 1924. Hamburg was an exciting place for someone with Schreier's interests and, among others, he was influenced by Emil Artin. The two worked together on the theory of knots and of braids.

Schreier married Edith Jakoby in 1928. He was a keen musician and played the piano and it was when he went for piano lessons in Hamburg that the two had met. Edith was older than Schreier, had been married before but her first husband had been killed in World War I. She had a fourteen year old son. A few months after he married, Schreier was offered a professorship at the University of Rostock in 1928 and decided to accept the position but he preferred to wait until the summer of 1929 before taking up the post. During the beginning of the 1928/29 session Schreier lectured on function theory giving parallel courses in Hamburg and Rostock. However, around Christmas of 1928, an illness which had been steadily worsening prevented him from continuing with his lectures. He died five months later at the age of 28 of a 'general sepsis'. The sulpha drugs discovered a few years later probably would have saved his life and therefore would have greatly changed the development of combinatorial group theory.

Although, as we have noted, many mathematicians influenced Schreier, the first were Furtwängler and Reidemeister. His first paper in 1924 *On the groups* *A*^{a}*B*^{b} = 1 gave a simple algebraic proof of a theorem on knot groups, which generalised a theorem given by Max Dehn ten years earlier that the trefoil knot and its mirror image are not equivalent. 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. He published the results in 1927 in the paper *Die Untergruppen der freien Gruppe* Ⓣ which is described in [5] as:-

... one of the most important papers ever published on combinatorial group theory. It took a long time for all its aspects to become effective, and it contains much more than the title indicates.

In January 1926 Schreier attended a lecture given by Reidemeister in Hamburg on finding presentations for normal subgroups of finitely presented groups. Reidemeister published his method later in 1926. Schreier, who took a more algebraic approach compared to Reidemeister's geometrical approach, 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. Schreier published his method in his 1927 paper *Die Untergruppen der freien Gruppe* Ⓣ. Other work of Schreier is described in [5] as follows:-

... Schreier made important contributions to other parts of group theory. The classical Lie groups ... can be considered as topological spaces. 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,39years after the publication of Hölder's paper. It is rare that such a widely used and basic theorem can be deepened after such a long time.(In this case, something even more unusual happened. Zassenhaus(1934)discovered a second improvement of the theorem.)

Finally let us look at Schreier's famous textbook. This arose from the lectures that he gave in Hamburg. 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. *Einführung in die analytische Geometrie und Algebra* Ⓣ was published in two volumes, the first in 1931 and the second in 1935. Henry George Forder, reviewing the first volume writes [6]:-

Schreier intended eventually to publish his Hamburg lectures on analytical geometry and algebra in book form; and after his premature death in1929these were completed and edited by his pupil Sperner. Three-quarters of the present first volume discuss the analytical geometry of n dimensions and the relevant algebra in a unified way; no special knowledge is assumed, but considerable demands are often made on the attention of the reader. ... The second part, on the theory of fields, begins by pointing out that in the earlier portion the basic symbols need not represent real numbers, but may stand for any elements satisfying certain laws of combination. These laws are the postulates which define a "field". The investigation of the general laws of divisibility and of the factoring of polynomials over any field, and the usual introduction to ordinary complex numbers, lead up to an interesting proof of the fundamental theorem of algebra in the ordinary complex field, and with this the volume practically ends. The remarkable developments in algebra in the last few years, in which the Hamburg school has played a notable part, are not treated at all in this book, but here and there, as in the use of the word "ideal" to denote certain sets of polynomials, and in some of the proofs, their influence is seen.

Reviewing the second volume, Forder writes [7]:-

This final volume contains an introduction to the theory of(finite)matrices, which has already been published separately and is now out of print, together with a treatment of n-dimensional analytic geometry from the projective, affine and metric points of view, as far as the classification of quadrics. The work on matrices is very thorough and detailed and uses methods of proof that have been developed comparatively recently...

An English translation appeared in 1951 under the title *Introduction to Modern Algebra and Matrix Theory*. Kurt Hirsch writes [9]:-

Otto Schreier's "Einführung in die Algebra und Analytische Geometrie"Ⓣ (edited by E Sperner)has for many years been a favourite text-book for first year undergraduates in German universities. One of its unique features is the simultaneous treatment of fundamental concepts in algebra and in affine and projective geometry. The present skilful English translation has incorporated the "Vorlesungen über Matrizen"Ⓣ (as in the German second edition), but has omitted the long final chapter on projective geometry in n dimensions. The result is that the accent is much more on the algebraic developments than in the original and that the geometry serves more as illustration than as motivation. This is also expressed in the changed title from which the reference to geometry has disappeared. ... The book can be thoroughly recommended both for private study and as a text-book for an enterprising Honours course combining algebra with geometry.

T C Holyoke of Northwestern University writes in the review [10]:-

This outstanding book forms a good introduction to some of the topics and methods of modern algebra, and appears to be well suited for use as a text in a course of that nature. The treatment of the subject matter is concrete, and frequent geometric illustrations and motivations are given. The concepts of vector and linear transformation underlie and motivate the discussion throughout. The translation covers volume one and half of volume two of the1935German edition(which differs considerably from the1948-51edition), omitting the chapter on projective geometry.

The part of the German text which was omitted from this English translation was published in 1961 as a separate book entitled *Projective Geometry of n Dimensions*. A J Knight writes [12]:-

Under the title of "Introduction to Modem Algebra and Matrix Theory," the Chelsea company has already published a translation of part of Schreier and Sperner's "Einführung in die Analytische Geometrie und Algebra." The present volume forms a companion to this and has the same source. It is a nicely produced book, written in a pleasant style and incorporating the formality of carefully stated theorems in an atmosphere which is occasionally almost conversational. Of the twelve chapters, the first seven are devoted to projective spaces, co-ordinates, cross-ratio, projectivities, duality, and correlations. The remaining chapters contain a discussion of quadrics and their classification in projective, affine and metric spaces. Where necessary, the cases in which the number field is real or complex are contrasted and at the end of each chapter a number of illuminating examples is provided. The treatment of projectivities is particularly interesting and, not-withstanding an unfortunate impression, given in the opening paragraph, that the main purpose of projective geometry is to eliminate the distinction between intersecting and parallel lines, the book can be recommended as a rigorous introduction to n-dimensional geometry for second-year university students.

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. Their anti-Semitic policies had a major impact on the Jewish Schreier family. Anna, Otto's mother, died in the Theresienstadt concentration camp in October 1942 and Theodor, Otto's father, died in the same concentration camp in January 1943. At least two of Otto's uncles were also murdered by the Nazis.

**Article by:** *J J O'Connor* and *E F Robertson*