Interconnections between the structure theory of set addition and rewritability in groups
Authors:
Gregory A. Freiman and Boris M. Schein
Journal:
Proc. Amer. Math. Soc. 113 (1991), 899910
MSC:
Primary 20F99; Secondary 20F05
MathSciNet review:
1072338
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Abstract: An approach to groups and semigroups stemming from the structure theory of set addition turns out to have much in common with the socalled permutation or rewritable properties. We explain these connections and show how these properties take their place in a wider class of interesting and naturally arising problems. As an example, we characterize some classes of groups and group elements.
 [1]
Russell
D. Blyth and Derek
J. S. Robinson, Recent progress on rewritability in groups,
Group theory (Singapore, 1987) de Gruyter, Berlin, 1989,
pp. 77–85. MR 981835
(90b:20035)
 [2]
L.
V. Brailovsky and G.
A. Freĭman, On twoelement subsets in groups, Fourth
international conference on collective phenomena (Moscow, 1981), Ann. New
York Acad. Sci., vol. 373, New York Acad. Sci., New York, 1981,
pp. 183–190. MR 719039
(84k:20003)
 [3]
L.
V. Brailovsky and G.
A. Freiman, On a product of finite subsets in a torsionfree
group, J. Algebra 130 (1990), no. 2,
462–476. MR 1051314
(91c:20052), http://dx.doi.org/10.1016/00218693(90)900934
 [4]
L.
Brailovsky, G.
Freiman, and M.
Herzog, Special elements in groups, Proceedings of the Second
International Group Theory Conference (Bressanone, 1989), 1990,
pp. 33–42. MR 1068347
(91g:20035)
 [5]
Mario
Curzio, Patrizia
Longobardi, and Mercede
Maj, On a combinatorial problem in group theory, Atti Accad.
Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 74 (1983),
no. 3, 136–142 (Italian, with English summary). MR 739397
(85k:20119)
 [6]
G.
A. Freĭman, Foundations of a structural theory of set
addition, American Mathematical Society, Providence, R. I., 1973.
Translated from the Russian; Translations of Mathematical Monographs, Vol
37. MR
0360496 (50 #12944)
 [7]
, Groups and the inverse problems of the additive set theory, NumberTheoretic Investigations of the Markov Spectrum and the Structure Theory of Set Addition, Kalinin Univ., Moscow, 1973, pp. 175183. (Russian)
 [8]
G.
A. Freĭman, On two and threeelement subsets of
groups, Aequationes Math. 22 (1981), no. 23,
140–152. MR
645414 (83e:20029), http://dx.doi.org/10.1007/BF02190175
 [9]
, What is the structure of if is small? Lecture Notes in Math., vol. 1240, SpringerVerlag, New York, 1987, pp. 109134.
 [10]
Gregory
A. Freiman and Boris
M. Schein, Group and semigroup theoretic considerations inspired by
inverse problems of the additive number theory, Semigroups, theory and
applications (Oberwolfach, 1986) Lecture Notes in Math., vol. 1320,
Springer, Berlin, 1988, pp. 121–140. MR 957761
(89j:20040), http://dx.doi.org/10.1007/BFb0083425
 [1]
 R. D. Blyth and D. J. S. Robinson, Recent progress on rewritability in groups, Group Theory (Proc. of the 1987 Singapore Conf.), de Gruyter, Berlin, 1988. MR 981835 (90b:20035)
 [2]
 L. V. Brailovsky and G. A. Freiman, On twoelement subsets in groups, Ann. New York Acad. Sci., vol. 373, 1981, New York Acad. Sci., New York, pp. 183190. MR 719039 (84k:20003)
 [3]
 , On a product of finite subsets in a torsionfree group, J. Algebra 130 (1990), 462476. MR 1051314 (91c:20052)
 [4]
 L. V. Brailovsky, G. A. Freiman, and M. Herzog, Special elements in groups, Group Theory (Proc. 2nd Internat. Conf., Bressanone, Italy 1989), Suppl. Rend. Circ. Mat. Palermo, II. Ser. 23 (1990), 3342. MR 1068347 (91g:20035)
 [5]
 M. Curzio, P. Longobardi, and M. Maj, Su di un problema combinatorio in teoria dei gruppi, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 74 (1983), 136142. MR 739397 (85k:20119)
 [6]
 G. A. Freiman, Foundations of a structural theory of set addition, Kazan, 1966 (Russian); English transi.: Trans. Math. Monographs, Amer. Math. Soc., Providence, RI, 1973. MR 0360496 (50:12944)
 [7]
 , Groups and the inverse problems of the additive set theory, NumberTheoretic Investigations of the Markov Spectrum and the Structure Theory of Set Addition, Kalinin Univ., Moscow, 1973, pp. 175183. (Russian)
 [8]
 , On twoand threeelement subsets of groups, Aequationes Math. 22 (1981), 140152. MR 645414 (83e:20029)
 [9]
 , What is the structure of if is small? Lecture Notes in Math., vol. 1240, SpringerVerlag, New York, 1987, pp. 109134.
 [10]
 G. A. Freiman and B. M. Schein, Group and semigroup theoretic considerations inspired by inverse problems of the additive number theory, Lecture Notes in Math., vol. 1320, SpringerVerlag, New York, 1988, pp. 121140. MR 957761 (89j:20040)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199110723383
PII:
S 00029939(1991)10723383
Article copyright:
© Copyright 1991
American Mathematical Society
