Questions in algebra and mathematical logic. Scientific heritage of S. I. Adian

This is a survey of results on the Burnside problem and properties of Burnside groups, the finite basis problem for group identities, periodic products of groups and Malcev’s problem, construction of groups with special properties (Tarski monsters), constructive bounds in the Burnside- Magnus problem, and algorithmic problems: the problem of recognition of group properties, the word problem for semigroups with one relation, and semi-Thue systems. The focus is on the most important results obtained in papers of Adian and his students. Bibliography: 81 titles.

[1]  Dieter Hofbauer,et al.  Termination of {aa->bc, bb->ac, cc->ab} , 2006, Inf. Process. Lett..

[2]  Sergei Ivanovich Adian,et al.  On Bounded Generation of SLN(Z) , 1992, Int. J. Algebra Comput..

[3]  S. I. Adian.  On a method for proving exact bounds on derivational complexity in thue systems , 2012 .

[4]  Michael O. Rabin,et al.  Recursive Unsolvability of Group Theoretic Problems , 1958 .

[5]  A. Ol'shanskii.  ON THE NOVIKOV-ADYAN THEOREM , 1983 .

[6]  S. Ivanov,et al.  Hyperbolic groups and their quotients of bounded exponents , 1996 .

[7]  Sergei Ivanovich Adian,et al.  On a semicontinuous function , 2011 .

[8]  J. G. Mennicke,et al.  Defining Relations and the Algebraic Structure of the Group SL2 over Integral Hamilton Quaternions , 1997, Int. J. Algebra Comput..

[9]  D. J. Collins,et al.  Review: Ju. V. Matijasevic, Sue Walker, Simple Examples of Undecidable Associative Calculi , 1968 .

[10]  A. Olshanskii.  Groups of bounded period with subgroups of prime order , 1982 .

[11]  Marshall Hall,et al.  Solution of the Burnside problem for exponent six , 1958 .

[12]  W. Magnus.  Das Identitätsproblem für Gruppen mit einer definierenden Relation , 1932 .

[13]  S. Adjan,et al.  Defining Relations and Algorithmic Problems for Groups and Semigroups , 1967 .

[14]  Sergei Ivanovich Adian,et al.  The Burnside problem and related topics , 2011 .

[15]  S. Gersten.  Essays in Group Theory , 2011 .

[16]  T. Delzant,et al.  Courbure mésoscopique et théorie de la toute petite simplification , 2008 .

[17]  G. Higman.  Subgroups of finitely presented groups , 1961, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[18]  P. Novikov,et al.  DEFINING RELATIONS AND THE WORD PROBLEM FOR FREE PERIODIC GROUPS OF ODD ORDER , 1968 .

[19]  A. Ol'šanskiĭ.  ON THE PROBLEM OF A FINITE BASIS OF IDENTITIES IN GROUPS , 1970 .

[20]  S. I. Adyan.  Transformations of words in a semigroup presented by a system of defining relations , 1976 .

[21]  M. Vaughan-Lee.  Uncountably many Varieties of Groups , 1970 .

[22]  koltsakh Li,et al.  E. I. Zel’manov Solution of the restricted Burnside problem for groups of odd exponent , 2018 .

[23]  S. Adjan.  ON SOME TORSION-FREE GROUPS , 1971 .

[24]  P. Novikov,et al.  INFINITE PERIODIC GROUPS. I , 1968 .

[25]  W. Magnus,et al.  Varieties of Groups , 1982 .

[26]  Paul E. Schupp,et al.  Embeddings into hopfian groups , 1971 .

[27]  S. I. Adyan,et al.  Lower estimates of the order of maximal periodic groups of prime period , 1988 .

[28]  S. I. Adyan,et al.  ON GROUPS ALL OF WHOSE PROPER SUBGROUPS ARE FINITE CYCLIC , 1992 .

[29]  SERGEI V. IVANOV,et al.  The Free Burnside Groups of Sufficiently Large exponents , 1994, Int. J. Algebra Comput..

[30]  P. Novikov,et al.  ON ABELIAN SUBGROUPS AND THE CONJUGACY PROBLEM IN FREE PERIODIC GROUPS OF ODD ORDER , 1968 .

[31]  Errata: A Connection between the Baker-Hausdorff formula and a Problem of Burnside , 1950 .

[32]  I. Lysenok.  Infinite Burnside groups of even exponent , 1996 .

[33]  A. Markov,et al.  Insolubility of the Problem of Homeomorphy ∗ , 2001 .

[34]  A. O. Houcine.  On hyperbolic groups , 2006 .

[35]  S. I. Adian,et al.  Periodic groups and Lie algebras , 1987 .

[36]  S. I. Adian.  New estimates of odd exponents of infinite Burnside groups , 2015 .

[37]  Rémi Coulon,et al.  On the geometry of burnside quotients of torsion free hyperbolic groups , 2014, Int. J. Algebra Comput..

[38]  Sheila Oates,et al.  Identical Relations in Groups , 1963 .

[39]  Once more on periodic products of groups and on a problem of A. I. Mal’tsev , 2010 .

[40]  S. I. Adian,et al.  The Hopfian property of n-periodic products of groups , 2015, 1504.02984.

[41]  Enno Ohlebusch,et al.  Term Rewriting Systems , 2002 .

[42]  Sergei Ivanovich Adian,et al.  The Burnside problem and identities in groups , 1979 .

[43]  J. Müller,et al.  Group Theory , 2019, Computers, Rigidity, and Moduli.

[44]  H. E. Rose.  Subrecursion: Functions and Hierarchies , 1984 .

[45]  H. Zassenhaus.  Über Lie’sche Ringe mit Primzahlcharakteristik , 1939 .

[46]  Yuri V. Matiyasevich,et al.  Decision problems for semi-Thue systems with a few rules , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[47]  E. Zelmanov.  SOLUTION OF THE RESTRICTED BURNSIDE PROBLEM FOR GROUPS OF ODD EXPONENT , 1991 .

[48]  H. Rice.  Classes of recursively enumerable sets and their decision problems , 1953 .

[49]  Exponential lower estimate of the degree of nilpotency of Engel Lie algebras , 1986 .

[50]  G. E. Wall.  On the Lie Ring of a Group of Prime Exponent , 1978 .

[51]  E I Zel'manov,et al.  A SOLUTION OF THE RESTRICTED BURNSIDE PROBLEM FOR 2-GROUPS , 1992 .

[52]  S. I. Adyan,et al.  Problems of equality and divisibility in semigroups with a single defining relation , 1987 .

[53]  M. Stanko.  On the Markov theorem on algorithmic nonrecognizability of manifolds , 2007 .

[54]  S. Adjan.  INFINITE IRREDUCIBLE SYSTEMS OF GROUP IDENTITIES , 1970 .