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. At Berlin he was taught by some outstanding mathematical teachers including Weierstrass, Kummer and Kronecker. He took the examinations to become a secondary school teacher in 1878 but he was aiming higher than this for he wanted to become a university teacher. In common with the standard practice of the time he moved around different German universities, next going to Göttingen from where he received his doctorate in 1880. Realising that it would be almost impossible to obtain a university position, Maschke decided to take up secondary school teaching.
His first teaching post was in the Luisenstädtische Gymnasium in Berlin. However he found both the long hours teaching and the elementary nature of the mathematics he was having to teach made him feel that he was in the wrong profession. This said, he was by all accounts a very good school teacher. He was given a sabbatical year and returned to Göttingen for the year 1886-87, working there with Klein. He had become a close friend of Bolza's while the two studied together at Berlin, and they were now together again at Göttingen both working with Klein. 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 . Hermite, Kronecker and Brioschi had, in 1858, discovered how to solve the quintic equation by means of elliptic functions. 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. However, he wrote to Klein on 8 December 1888:-
... everyone here works in isolation and can hardly be moved to talk about his research.By 1889 he had made a definite decision to give up school teaching. In this year Bolza, who was by now in the United States, was working at Johns Hopkins University where he had been given a temporary short-term appointment, and he had already accepted an appointment at Clark University in Worcester, Massachusetts. 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. Keeping his teaching post, he began part-time study of electrotechnics at the Polytechnicum in Charlottenburg in 1889-90. In 1890 he resigned his teaching post and took up full-time technical training in Darmstadt.
In 1891 Maschke emigrated to the United States and worked for a year with the Western Electrical Instrument Company, Newark, New Jersey. In 1892 the University of Chicago opened and the head of the mathematics department, Eliakim Moore, began building up a strong unit. Bolza joined the University of Chicago in 1892 and then he persuaded Moore to appoint Maschke to Chicago. The three were highly influential in building up a strong mathematics research school in Chicago. R C Archibald writes:-
These three men supplemented one another remarkably. 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. During the period 1892-1908 the University of Chicago was unsurpassed in America as an institution for the study of higher mathematics.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 G is not divisible by the characteristic of the field K, then the (finite-dimensional) K-representations of G are completely reducible. A closer look at events surrounding this provide an interesting insight into the interplay between various mathematicians. Maschke proved a special case of his theorem in the paper Über den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen published in 1898. The general result appeared in the following year in Beweiss des Satzes, dass diejenigen endlichen linearen Substitutionesgruppen, in welchen einige durchgehends verschwindende Coefficienten auftenen intransitiv sind . 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. He had subsequently written it up in a paper which he submitted to Klein for publication in Mathematische Annalen. Klein reported to Moore that Alfred Loewy had stated the result without proof in an article he published in 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. In this area, which he started to work in after 1900, he led the symbolic treatment of the subject.
At Chicago, together with Moore, Maschke was responsible for the rapid rise to eminence of the University in mathematics research. David Smith writing in  says:-
He devoted the remainder of his life to the training of mathematicians and to assisting in building up and maintaining a strong department in that university. He was a teacher of great ability and his courses were made more valuable by his all-round culture, by his originality of thought, and by his personal interest in the large numbers of young mathematicians who attended his lectures.Among the papers he published while at Chicago are: On systems of six points lying in three ways in involution (1896), Note on the unilateral surface of Möbius (1900), A new method of determining the differential parameters and invariants of quadratic differential quantics (1900), On superosculating quadric surfaces (1902), A symbolic treatment of the theory of invariants of quadratic differential quantics of n variables (1903), Differential parameters of the first order (1906); The Kronecker-Gaussian curvature of hyperspace (1906) and A geometrical problem connected with the continuation of a power-series (1906). 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.
Eliakim Moore recognised the influence of Klein, through Bolza and Maschke, on his leading American research university and wrote to him in 1904 saying :-
Certainly in the domain of mathematics German scholars in general and yourself in particular have played, by way of example and counsel and direct and indirect inspiration, quite a leading role in the development of creative mathematicians in this country ...At the end of February 1908 Maschke entered hospital to undergo emergency surgery. He died following complications.
Article by: J J O'Connor and E F Robertson