**Heinrich Maschke** was a German mathematician born in what is now Poland who worked on group theory and on differential geometry.

- Heinrich attended the Gymnasium in Breslau where he showed great ability.
- Military service was required at that time so Maschke spent a year in the army before he continued his studies at the University of Berlin.
- Realising that it would be almost impossible to obtain a university position, Maschke decided to take up secondary school teaching.
- Maschke found working with Klein in his home in the evenings very rewarding and was fascinated with Klein's ideas on using group theory to solve algebraic equations.
- Due to Klein's encouragement, Maschke published his first paper in 1887, namely Über die quaternäre endliche, lineare Substitutionsgruppe der Borchardt'schen Moduln Ⓣ(On the quaternary finite, linear substitution group of modules of Borchardt).
- In 1888 Maschke proved that a particular sixth-degree equation could be solved by using hyperelliptic functions and Brioschi showed that any sixth-degree algebraic equation could be reduced to Maschke's equation and therefore solved in the same way.
- Maschke had returned to the Gymnasium in Berlin before he wrote the paper we just mentioned but, as is evident, he was not teaching in a secondary school yet concentrating on research.
- Maschke decided that the best course of action for him was to follow Bolza to the United States but Bolza warned him that it was not easy to get academic positions there.
- Maschke thought he had better have qualifications to enter some other profession otherwise he might emigrate to the United States and end up there as a school teacher - the profession he had now decided to give up.
- In 1891 Maschke emigrated to the United States and worked for a year with the Western Electrical Instrument Company, Newark, New Jersey.
- Bolza joined the University of Chicago in 1892 and then he persuaded Moore to appoint Maschke to Chicago.
- Moore was a fiery enthusiast, brilliant, and keenly interested in the popular mathematical research movements of the day; Bolza, a product of the meticulous German school of analysis led by Weierstrass, was an able, and widely read research scholar; Maschke was more deliberate than the other two, sagacious, brilliant in research, and a most delightful lecturer in geometry.
- Between 1892 and 1910 the mathematics department was outstandingly successful with thirty-nine students graduating with doctorates (but only five of them were students of Maschke).
- Maschke was promoted to associate professor in 1896 and then to full professor in 1907.
- Under Klein's inspiration while at Göttingen, Maschke had worked in group theory, in particular working on finite groups of linear transformations.
- He is best known today for Maschke's theorem, which he published in 1899, which states that if the order of the finite group GGG is not divisible by the characteristic of the field KKK, then the (finite-dimensional) KKK-representations of GGG are completely reducible.
- Maschke proved a special case of his theorem in the paper Über den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen Ⓣ(On the arithmetic nature of the coefficients of linear finite substitution groups) published in 1898.
- In his proof Maschke used a theorem by Moore which he had announced to the Mathematics Club at the University of Chicago on 10 July 1896.
- Moore's paper appeared in Mathematische Annalen two years later and the Loewy-Moore theorem provided Maschke with a critical step in the proof of his own theorem.
- Maschke's second area of work was on differential geometry in particular the theory of quadratic differential quantics.
- At Chicago, together with Moore, Maschke was responsible for the rapid rise to eminence of the University in mathematics research.
- Six of these papers were published in the Transactions of the American Mathematical Society and Maschke played a large role in getting the American Mathematical Society established, being a founder member of the Chicago Section of the Society in 1897.
- Maschke served on the Council of the American Mathematical Society from 1902 to 1905 and was vice president of the Society in 1907.
- At the end of February 1908 Maschke entered hospital to undergo emergency surgery.

Born 24 October 1853, Breslau, Prussia (now Wrocław, Poland). Died 1 March 1908, Chicago, Illinois, USA.

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Group Theory, Origin Poland

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