**Lev Arkad'evich Kaluznin** was a Russian mathematician who is best known for his work in group theory and in particular permutation groups.

- She had a great influence on Lev Arkad'evich throughout her life - sometimes markedly so.
- In 1925 Lev entered a secondary school (Realschule) of high academic standing, from which he graduated in 1933.
- While at Humboldt he was greatly influenced by Schur, whose lectures on algebra shaped Lev's mathematical interest.
- On 22 June 1941 his life changed dramatically - as did the lives of many Soviet citizens, who, like Kaluznin, were interned and sent to a camp in Compiègne near Paris.
- (In fact, in later years, Lev Arkad'evich could still recall great lectures he had heard there on world history, Roman law, etc.) During this time he did some research in Galois theory.
- In the Spring of 1945 Kaluznin returned to Paris.
- To meet this condition Kaluznin began working at the Humboldt University in Berlin in 1951 - first as a lecturer, and then later, after habilitation with his thesis Stable automorphism groups, as a full professor.
- In 1955 Lev Arkad'evich returned to the USSR.
- In 1957 Kaluznin defended his postdoctoral thesis (a Soviet version of habilitation) on the topic Sylow p-subgroups of symmetric groups.
- Having always been perceived as an "alien", Kaluznin was forced to leave his position as Head of the Department of Algebra and Mathematical Logic, though he retained his professorship at Kiev State University until 1985.
- In 1984, due to deteriorating health, Lev Arkad'evich relinquished his teaching duties, and in 1985 his position within the faculty was changed to that of a 'senior researcher'.
- All this contributed heavily to Kaluznin's forced retirement and to his subsequent move to Moscow.
- As time passed Kaluznin's health deteriorated, and his death came as the result of severe burns caused by an accident.
- As a researcher, Kaluznin is best known for his work in group theory and in particular permutation groups.
- Despite the fact that the earliest applications of wreath products of permutation groups was due to C Jordan, W Specht and G Polya, it was Kaluznin who first developed special computational tools for this purpose.
- A particularly important result is the well-known theorem of Krasner and Kaluznin concerning the embeddings of a group with a subnormal series into the wreath product of the factors of the series.
- Kaluznin made several applications of the wreath product to mathematical logic and mathematical chemistry.
- Kaluznin's other significant contributions to group theory include his work on stable automorphism groups, the structure of the variety of nnn-abelian groups, a classification of metabelian groups, work on locally normal groups of higher categories, and characterisations of the maximal subgroups of the symmetric and alternating groups.
- Another area of algebra which had always attracted Kaluznin's interest was Galois theory.
- Developing further the methods of abstract Galois theory which had been initiated by Krasner, Kaluznin and his students were able to establish a Galois correspondence between Post algebras and Krasner algebras.
- Kaluznin also worked in the area of geometrical algebra, particularly on arrangements of subspaces in euclidean and unitary spaces.
- Though Lev Arkad'evich had never considered himself to be an expert in mathematical linguistics, automata theory, and applications of computers in algebra, his mathematical interests were broad and he did not hesitate to do research in these areas outside of his main strengths.
- Kaluznin was an outstanding teacher.
- Lev Arkad'evich had a gift for recognising sparks of talent and potential in his students at a very early stage, often during their first semester at Kiev State University.
- Despite the absurdities and humiliations of Soviet life, with its constant dependencies on the whims of those who were in power, Lev Arkad'evich was never heard to regret his return to the USSR.
- Lev Arkad'evich used to be a heavy smoker, smoking sixty a day even during his lectures.
- Kaluznin was associated with many members of Kiev's intellectual elite.
- Lev Arkad'evich liked and valued good red wines and also good beer.
- This unconventional attire prompted the joke among his students that Kaluznin is the "only real doctor of the Department".
- On the other hand, Kaluznin sometimes found it difficult to part with old clothing.
- To the secretary's dismay, she later saw a smiling Lev Arkad'evich, jacket in hand, puzzled by how it came to be in the bin.

Born 31 January 1914, Moscow, Russia. Died 6 December 1990, Moscow, Russia.

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Group Theory, Origin Russia

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