**Aleksandr Kurosh** proved important results in Group Theory and is best-known as the author of one of the standard text-books in the subject.

- In the middle of the war, in 1916, Shura entered school, going straight into the third class.
- Kurosh left school at the age of 15 and went to Moscow to take the examinations for the Textile Institute.
- He achieved very high grade passes in the entrance examinations but was judged to be too young to enter the Institute.
- Kurosh never intended to spend his life as an accountant and he studied at evening classes after a full working day.
- Up to this time, unlike many great mathematicians, he had no particular passion for the subject.
- His special topic at the evening classes was the stream engine.
- In 1924 Kurosh became a student at Smolensk University.
- At that time Emmy Noether was in Moscow and gave a course at Moscow State University on abstract algebra, which Aleksandrov attended.
- Under the fresh impression of these lectures he gave a course on modern algebra at Smolensk.
- In 1929 Kurosh was assigned to the Institute of Mathematics and Mechanics at Moscow State University.
- This was done at Aleksandrov's request since he was by this time in Moscow and wished to continue to supervise Kurosh's work.
- However, although Kurosh's first results were in topology, solving problems posed by Aleksandrov, he was already interested in the theory of groups.
- He had read O Yu Schmidt's group theory papers while still in Smolensk so when he found himself able to attend Schmidt's seminar at Moscow State University, his interest in groups increased further.
- However, after attending Schmidt's group theory course in 1930, he found himself taking over some of Schmidt's duties when he left the university in the autumn of that year.
- Kurosh was appointed an assistant at Moscow State University in 1930 becoming a lecturer in 1932 and a professor there in 1937.
- The professorial appointment came after Kurosh was awarded his doctorate for a thesis entitled Research on infinite groups which he defended on 22 April 1936.
- It should be mentioned that the doctorate was a much higher degree than the present British/American PhD which compares in level with the Master's Degree in Russia.
- In 1949 he became a director at Moscow State University where he continued to work throughout his career.
- The Chair of Higher Algebra at Moscow State University was created in 1929 and O Yu Schmidt was the first occupant of the Chair.
- Kurosh was the second holder of the Chair, holding the Chair from 1949 until his death in 1971.
- Like most Russian academics of that period, Kurosh had a variety of other attachments teaching courses and lectured at a number of other institutions in Moscow.
- As mentioned above, Kurosh's first significant results were in topology, solving a problem set by Aleksandrov.
- Quickly he moved to research in group theory and his first paper on this topic appeared in 1932 on direct decompositions of groups.
- Shortly after completing this work, Kurosh came across Schreier's papers on free products of groups.
- Soon he was producing important results on the topic and two papers came out of his work on free products.
- The second of these papers appeared in Mathematische Annalen and contains a proof of the celebrated Kurosh subgroup theorem, which describes subgroups of a free product of groups.
- This paper brought Kurosh international fame.
- Kurosh is best known for his book The Theory of Groups which was written in two volumes.
- In 1952 Kurosh brought out a second edition of the book which was almost a new book given that it attempted to cover the large amount of progress during the years 1940-52.
- The book includes many of Kurosh's own results on groups, in particular the Kurosh Subgroup Theorem mentioned above.
- However, Kurosh did not spend all his research efforts on group theory.
- Some of this work was described in his other famous textbook Lectures on General Algebra published in 1960 which became an internationally famous text.
- As with The Theory of Groups this text was also translated into English by Hirsch.
- During the 1950s he concentrated on Universal algebra and category theory, organising a major seminar on category theory in 1958.
- Many mathematicians participated in this seminar and it led to the birth of the Moscow School of Category Theory.
- between the theory of universal algebras and the classical branches of general algebra there exists a big uncultivated space.
- Research has begun only in a few places, isolated, sometimes random ...
- One has to expect that it is precisely in this no-man's land where the basic branches of general algebra will move in the next decades.
- In accordance with the general tendencies of contemporary science ...
- Kurosh's whole life was involved in teaching at all levels from directing the research of young people, to lecturing, to designing new courses.
- Twice he organised the Moscow University Olympiads for school mathematics.
- One way was his long involvement with the Moscow Mathematical Society.
- In 1931, while he was still an assistant, Kurosh joined the Moscow Mathematical Society and he was elected to its governing body in 1933, only a year after his appointment as a lecturer.
- In that year he gave his first lecture to the Society on Fundamental trends in finite group theory.
- During the years 1943-48 Kurosh served the Society as its Librarian and then for 22 years from 1948 he served continuously on the governing body of the Society.
- During six of these years he was Vice-President.

Born 19 January 1908, Yartsevo (near Smolensk), Russia. Died 18 May 1971, Moscow, Russia.

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Group Theory, Origin Russia

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive