**Lazar Matveevich Gluskin**was born in the city of Bakhmut but in 1924, two years after he was born, the town was renamed Artemovsk. It is now in Donetsk province, eastern Ukraine, on the Bakhmut River. However, Donetsk was formerly (until 1924, so for the first two years of Gluskin's life) Yuzovka, then (1924-61) Stalino. Lazar Matveevich was born into a Jewish family. In 1939, when he was seventeen years old, he entered the Physics and Mathematics Faculty of Kharkov University. He studied there unaffected by World War II until 1941 when German troops turned on their Soviet ally and invaded the Soviet Union on 22 June.

Operation Barbarossa, as the invasion of the Soviet Union was called by the Germans, was intended to achieve a rapid victory, and indeed the progress of the German armies across a broad front stretching from the Baltic to the Black Sea was at first remarkable. By September Kiev had fallen and large numbers of Soviet troops surrendered [5]:-

But Gluskin was determined to study mathematics and returned to Kharkov University. There his studies were supervised by Anton Kazimirovic Suschkevic, one of the first mathematicians to make major progress in the theory of semigroups, and in 1949 Gluskin was awarded his Diploma in Mathematics. Although by now a great enthusiast for semigroups, Gluskin could not continue to undertake research at Kharkov University but was forced to take a teaching position at the Kharkov Pedagogical Institute. Despite a heavy teaching load, he undertook research in his own time, unofficially supervised by Suschkevic which was both a kind and brave act on his part. Life during these years was extremely hard [5]:-... German troops occupied Kharkov. Walking, riding in railway freight cars, hitchhiking, Gluskin managed to escape. He continued his studies at Saratov University. His father died; his mother tried to support her two children taking low-paid menial jobs, so he had to look for odd jobs, mainly unloading railway freight cars and barges on the Volga. In1943he was drafted into the Red Army and soon became a commander of an anti-aircraft battery fighting the German Luftwaffe and, after the war with Germany was over, fighting the Japanese Air Force in China. The war left him a veteran with numerous military decorations, but even his friends doubted he could ever resume his mathematical studies.

Guskin was awarded his Candidates Degree (equivalent to a Ph.D.) in 1952 from Kharkov University. He continued to teach at the Kharkov Pedagogical Institute but things became even worse on the accommodation front in 1956 when his second daughter Valentina was born. However, Kharkov was an important cultural centre and Gluskin was able to mix with many leading mathematicians there. It was therefore a hard decision to leave the city in 1958 and accept the chair of mathematics at the Institute of Mining and Metallurgy in Voroschilovsk. This small town had only one plus for Gluskin, and that was the fact that he was offered an appartment to rent by his employer, so his family's living conditions were vastly improved by the move.... Gluskin and his wife Tamara lived in a tiny cubicle in a student hostel, and when their first child Irina arrived he was finishing his dissertation using the only "office space" available: the staircase outside their room.

The first of Gluskin's papers *An associative system of square matrices* (Russian) appeared in 1954. This paper on matrix semigroups was one of a number that he published on this topic near the beginning of his career; *Automorphisms of multiplicative semigroups of matrix algebras* (Russian) (1956) was another on this topic. His other early papers included *Homomorphisms of unilaterally simple semigroups on groups* (Russian) (1955), *Simple semigroups with zero* (Russian) (1955), and *Elementary generalized groups* (Russian) (1957). He began to become interested in semigroups of continuous transformations of topological spaces publishing papers such as *The semi-group of homeomorphic mappings of an interval* (Russian) (1959) and *Automorphisms of semigroups of topological mappings* (Russian) (1960) on this topic.

In 1961 Gluskin's quality research was recognised when he was awarded a Doctor of Sciences degree from Moscow University. It was A G Kurosh, who held the Chair of Algebra at Moscow University, who organised that Gluskin defend his thesis there for the higher degree. The examiners, in addition to Kurosh, were E S Lyapin (Leningrad) and V V Vagner (Saratov). The award of the degree helped Gluskin to become a full professor at the Institute of Mining and Metallurgy in Voroschilovsk in 1962 and, three years later, to return to Kharkov to the chair of mathematics at the Kharkov Mining Institute. His health began to deteriorate and in 1968 S D Berman took over the chair while Gluskin continued to work as professor of mathematics relieved of the extra work and responsibilities of the chair.

Schein writes in [4]:-

For many years Gluskin worked on dense extensions of commutative semigroups. He wrote to Schein four months before his death [5]:-... he published a series of brilliant results on semigroups of linear transformations. He showed that a good deal of the classical theory of rings of linear transformations could be carried over to semigroups, addition not being really needed. Later he considered semigroups of endomorphisms of modules over rings, thus generalising results on semigroups of linear transformations. From his point of view the semigroups of transformations formed the real core of the theory of semigroups, thus justifying the existence of the whole theory.

In fact he completed writing up the paper, solved another one of two further problems he wanted to include but died before he completed the last one. He died at home, just as he was about to leave to go and teach.Finally, I solved the problem on maximal dense normal extensions of commutative semigroups. I announced this result once before, but there was an error. In the last fifteen years I returned to it again and again without success: technical details eluded me. I must confess it was vexing, for twice during these years I was standing with one foot in the next world, and I would rather not leave behind a published theorem which was not true.

In [4] Schein describes Gluskin as a supervisor of doctoral students:-

He was an enthusiast for attending conferences but restricted by the Soviet regime:-Gluskin supervised about twenty Ph.D. dissertations in algebra. Many more young people are grateful to him for his unselfish advice and help. The times were complicated, and many people had difficulties in publishing their results and getting academic degrees or jobs for reasons which had nothing to do with the quality of their work. If Gluskin was sure that the mathematical substance of the results was sound, he was there to help - even if this could hurt his own career. He could not live any other way.

As to his interests outside semigroup theory and mathematics, Schein paints the following picture:-Gluskin was very active at mathematical conferences. In thirty years he gave scores of talks at the conferences, colloquium lectures, and courses on invited lectures at universities. He had many invitations from foreign universities and mathematical conventions and wanted very much to attend, but he was never allowed to go abroad(if one does not count the visits he had to pay during the war). The only exception was his short visit to a semigroup conference in Czechoslovakia in June1968.

Gluskin's interest in and appreciation of the history of mathematics was a part of his general interest in history, ancient and new, in the development of human civilization and related moral questions. He loved books. If only he could find a good book, which could be a difficult if not impossible enterprise in the surrounding general atmosphere, he would read it and, if possible, recommend or lend it to his friends.

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