# Sergei Nikolaevich Chernikov

### Born: 11 May 1912 in Zagorsk, Russia

Died: 1987

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**Sergei Nikolaevich Chernikov**'s father was a priest. Chernikov did not have a research career in mind when he attended secondary school and he went through his school career with the aim of becoming a teacher of mathematics when he left school. Indeed that is precisely the route that he took, obtaining a post as a mathematics teacher in a secondary school. However by this time he did want to study mathematics further and so he registered as an external student at the Saratov Pedagogic Institute.

Having obtained a degree in mathematics from the Saratov Pedagogic Institute, Chernikov left school teaching and was appointed to the Faculty of the Ural Institute of Physics and Mechanics. By now he had become extremely interested in the recent developments in mathematics, and he began to study algebra reading works by D A Grave and Grave's pupils O J Schmidt, N G Chebotaryov. By 1936 Chernikov had no doubt that he wished to study group theory and so he applied to the University of Moscow to undertake research as an external student under Kurosh's supervision.

Chernikov was particularly interested in infinite groups and the way to attack problems in that area is to examine properties of classes of infinite groups which are in some sense close to finite. There are properties such as solubility, a concept which goes back to Galois and attempts to classify which polynomial equations could be solved by radicals, which make perfect sense for infinite groups. Much work has indeed been done on studying infinite soluble groups. The idea which Chernikov had, and today it seems so natural that it is hard to realise what a clever idea it was, was to study generalisations of properties such as 'soluble' which an infinite group could have but when restricted to finite groups reduced to the original concept. He also studied finiteness type conditions that had already been seen to have great importance in ring theory, namely finiteness type conditions which did not allow infinite chains of subgroups of a specified type.

Chernikov's first new results came early in his time as an external student of Kurosh. By 1938 he already had two papers published on generalisations of results from finite group theory to infinite group theory, in particular generalising Frobenius's theorem to infinite groups. His doctoral dissertation, which he defended in 1940, was on

*Infinite locally soluble groups*and this introduced generalisations of properties such as soluble and nilpotent to infinite groups of the type described above. To be precise a locally soluble group is one in which every finite collection of elements is contained in a soluble subgroup.

Even before the award of his doctorate, Chernikov had been made Head of the Mathematics Department at the Saratov Pedagogic Institute. Then in 1946 he was appointed as Head of the Faculty of Mathematics at the Ural State University. After five years at the Ural State University, Chernikov was appointed to a similar position at Perm University. The University of Perm had been founded in 1916, was called Molotov University for a time, and is now the Gorky State University. In 1961 Chernikov was appointed as Head of the Department of Algebra and Geometry of the Sverdlovsk branch of the Steklov Institute of the USSR Academy of Sciences.

In fact Chernikov had two special research interests, the first being that of infinite groups which we have referred to above. Together with Kurosh he wrote a survey article

*Soluble and nilpotent groups*in 1946. He then wrote a beautiful survey article

*Finiteness conditions in the general theory of groups*which was published in 1959 and contained many of Chernikov's own results and those of others. The authors of [2] and [3] write:-

In the middle 1960s I [EFR] began research on infinite groups and I made much use of the fine survey articles by Chernikov. One thing was clear, Chernikov was not just looking round for results which he could prove, he was developing a systematic theory in the way that is the hallmark of top quality mathematicians. The other algebraists mentioned in the quote above who began to help Chernikov in building his theory included O J Schmidt, Malcev, Baer, Kurosh, Hall, and others.Intensified and expanded both in papers by Chernikov himself and of his pupils ...and also in the works of other ... algebraists, the study of infinite groups with finiteness conditions, enriched group theory with many new concepts, ideas and profound results, and also widened considerably the basis of group theory, extending it by new detailed investigations of infinite groups of specific form.

We have still to discuss Chernikov's second research interest. This was the study of the theory of linear inequalities, an area of great practical significance because of its connection with the theory of linear programming. The authors of [6] and [7] write:-

In 1968 Chernikov wrote an important bookThe practical importance of convenient algorithms for the solution of systems of linear inequalities and their connection with the theory of linear programming is well known. In Chernikov's articles, therefore, a series of geometrically obvious properties of linear inequalities is given in analytic form that is more convenient for the use of machine techniques.

*Linear inequalities*which gives sets out Chernikov's algebraic theory. At ([2] and [3] ):-

A series of papers by Chernikov in the 1960s studied polyhedrally closed systems, special types of infinite systems of linear inequalities [2] and [3]:-... the basis of this theory lies in the principle of boundary solutions; all its results are deduced from it by means of only a few finite methods...

The reader will have noticed the parallel in Chernikov moving from finite to infinite systems of linear inequalities in a similar spirit to moving from finite to infinite groups.For the case of a finite-dimensional real linear space these are infinite systems whose adjoint cone is topologically closed. Some properties of finite systems of linear inequalities can be transferred to polyhedrally closed systems of linear inequalities. Polyhedrally closed systems of linear inequalities are an effective means in the analysis of problems of the theory of approximation of functions, in linear programming(in particular in questions of duality), and in control theory.

It is a pleasure to record here my [EFR] own personal thanks for Chernikov's work on finite conditions in infinite groups which inspired me in my own research and was the topic of my doctoral thesis.

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

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