# Hans Fitting

### Born: 13 November 1906 in München Gladbach (now Mönchengladbach), Germany

Died: 15 June 1938 in Königsberg, Prussia, Germany (now Kaliningrad, Russia)

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**Hans Fitting**'s father, Prof Dr Friedrich Fitting (1862-1945), was a graduate secondary school teacher and a research mathematician who published over twenty papers. His doctorate was from the Martin-Luther University of Halle-Wittenberg in 1888 for the thesis

*Über eine Klasse von Berührungstransformationen*Ⓣ. Albert Wangerin had been his thesis advisor. Friedrich Fitting is best known today for giving a proof, in 1931, that there are exactly 880 magic squares of order 4. This result appears in his paper

*Rein mathematische Behandlung des Problems der magischen Quadrate von 16 und von 64 Feldern*Ⓣ and is remarkable since these 880 magic squares had been given by Frenicle de Bessy in 1693 but no proof was found until Friedrich Fitting's 1931 paper appeared. He taught his son Hans in school and was thus able to quickly recognize his son's extraordinary mathematical talent. With his father's challenge to his understanding of the subject, Hans soon progressed far beyond what was customary for his age.

From 1925 to 1932 Fitting studied mathematics, physics and philosophy at the Universities of Tübingen and Göttingen, where he was awarded his Ph.D. in 1932 for his work on group theory. His thesis advisor at Göttingen was Emmy Noether. Wolfgang Krull and Emmy Noether had proposed the task of classifying the structure of automorphism rings in a general theory of generalized abelian groups, that is abelian groups with operators. Fitting achieved this in his dissertation

*Zur Theorie der Automorphismenringe Abelscher Gruppen und ihr Analogon bei nichtkommutativen Gruppen*Ⓣ. The oral examination was held on 29 July 1931, and the thesis was published in Berlin in 1932.

Following his Ph.D., Fitting continued his research at the Mathematical Institutes of the Universities of Göttingen and Leipzig, funded by the "Notgemeinschaft der Deutschen Wissenschaften" (which can be translated loosely as the Emergency Association for German Science). Emmy Noether advised Fitting to apply for this grant and she had strongly supported his application. In 1934 he published a paper which Emmy Noether greatly appreciated, namely

*Über die direkten Produktzerlegungen einer Gruppe in direkt unzerlegbare Faktoren*Ⓣ.

In April 1934 Fitting moved to Königsberg, where he worked as a research assistant and later as an assistant lecturer in mathematics. The position in Königsberg had arisen since, after the Nazis came to power in 1933, Jewish academics were forced out of their positions and Richard Brauer had been forced to leave Königsberg. Also Kurt Reidemeister, who had held a chair at Königsberg from 1927, had been forced to resign in 1933 because he was considered "politically unsound" by the Nazis. Gábor Szegő, also a professor of mathematics at Königsberg, was forced to leave during 1934. In fact it was through contacting Richard Brauer that Emmy Noether was able to secure the position for Fitting. She wrote a letter to Brauer thanking him for his help which was sent to the United States since by that time Brauer had left Königsberg. Fitting was not the only new mathematics appointment to Königsberg in April 1933 for Wilhelm Specht was appointed at the same time. Fitting habilitated at the Science Faculty of the University of Königsberg in April 1936 and on 1 November 1937 he was promoted to lecturer for mathematics. His habilitation thesis was published in two parts;

*Die Determmantenideale eines Moduls*Ⓣ (1936) and

*Der Normenbegriff für die Ideale eines Ringes beliebiger Struktur*Ⓣ (1937). In his brief spell as a lecturer, Fitting led classes on a wide range of mathematical topics and distinguished himself, in particular, by his great attention to detail.

As a mathematician Fitting followed a reflective, yet meticulous approach, which allowed him to fully explore his ideas and their consequences. As a result, he was able to reach many clear and fundamental conclusions in group theory.

Among the many mathematical achievements of Fitting we note that he gave a proof of the Remak-Krull-Schmidt theorem on the uniqueness of the direct product decomposition of groups into indecomposable subgroups, even for groups of operators. He devoted himself to an investigation of the ideal theory of noncommutative rings and also studied the theory of determinant ideals of finitely generated modules

*M*over a commutative ring

*R*. In his 33-page paper

*Die Determinantenideale eines Moduls*Ⓣ (1936), the first part of his habilitation thesis, Fitting introduced what today are called 'Fitting ideals' of

*M*. Also in this paper is the well-known 'Fitting's Lemma' which states that if

*A*and

*B*are two normal nilpotent subgroups of a group

*G*with classes

*a*and

*b*respectively, then

*AB*is a nilpotent subgroup of class at most

*a*+

*b*.

Today, as well as for Fitting's Lemma, he is remembered for the 'Fitting subgroup' which is used in the structure theory of finite groups: every finite group

*G*possesses a unique largest normal nilpotent subgroup, the Fitting subgroup

*F*(

*G*). Since the factor group

*G*/

*F*(

*G*) is non-trivial for finite groups

*G*that are not nilpotent, this gives rise to the Fitting length of a finite group.

For a list of all Fitting's publications, see THIS LINK.

On 6 June 1938, after a long illness, Fitting succumbed to bone cancer, aged only 31.

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

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**Mathematicians born in the same country**

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