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On automorphisms and embeddings of free periodic groups. (English. Russian original) Zbl 1299.20057
J. Contemp. Math. Anal., Armen. Acad. Sci. 46, No. 2, 106-112 (2011); translation from Izv. Nats. Akad. Nauk Armen., Mat. 46, No. 2, 59-70 (2011).
Summary: The paper gives a construction of a free monoid of rank 2 in the group of automorphisms of free periodic groups \(B(m,n)\) of any odd period \(n\geq 665\) and any rank \(m>1\). Moreover, it is proved that if the period is any prime number \(n>1003\) and the group \(B(m,n)\) is nested in some \(n\)-periodic group \(G\) as a normal subgroup, then \(B(m,n)\) is a direct factor in \(G\).
20F50 Periodic groups; locally finite groups
20F05 Generators, relations, and presentations of groups
20F28 Automorphism groups of groups
20M05 Free semigroups, generators and relations, word problems
20E07 Subgroup theorems; subgroup growth
Full Text: DOI
[1] S. I. Adian, The Burnside problem and identities in groups (Nauka, Moscow, 1975).
[2] P. S. Novikov, S. I. Adian, ”On infinite periodic groups. I, II, III”, Izv. AN SSSR, Ser.Matem., 32, 212–244, 251–524, 709–731 (1968).
[3] E. A. Cherepanov, ”Normal automorphisms of free Burnside groups of large odd exponents”, Intern., J. Algebra Comput., 16(5), 839–847 (2006). · Zbl 1115.20024 · doi:10.1142/S0218196706003268
[4] V. S. Atabekyan, ”Non \(\phi\)-admissible normal subgroups of free Burnside groups”, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), 45(2), 21–36 (2010). · Zbl 1299.20046
[5] V. S. Atabekyan, ”Normal automorphisms of free Burnside groups”, Izv. Rus. Akad. Nauk, Ser.Mat., 75(2), 3–18 (2011). · Zbl 1227.20030 · doi:10.4213/im4256
[6] E. A. Cherepanov, ”Free semigroup in the group of automorphisms of the free Burnside group”, Communications in Algebra, 33:2, 539–547 (2005). · Zbl 1121.20028 · doi:10.1081/AGB-200047435
[7] A. S. Pahlevanyan, ”Infinite order automorphisms of free periodic groups of sufficiently large exponent”, Proceedings of the Yerevan State University, Phys. and Math. Sciences, 219(2), 38–42 (2009). · Zbl 1351.20023
[8] A. S. Pahlevanyan, ”On group of automorphisms of free periodic groups”, in: 210–214, Anniversary Scientific Conference Devoted to 90 Anniversary of EGU, 1 (YSU Press, Yerevan, 2009).
[9] A. S. Pahlevanyan, ”Free subgroup of group of automorphisms of free Burnside group”, in: p. 72, International Conference ”Maltsev Readings” Devoted to the Centenary of Anatoly Ivanovich Maltsev (NSU Press, 2009). · Zbl 1351.20023
[10] Rémi Coulon, ”Outer automorphisms of free Burnside groups”, arXiv:1008.4495, 1–19 (2010). · Zbl 1216.20032
[11] Rémi Coulon, Automorphismes exterieurs du groupe de Burnside libre (PhD Thesis, Universite de Strasbourg, 2010). · Zbl 1216.20032
[12] Derek J. S. Robinson, A course in the theory of groups. Graduate Texts in Mathematics, 80 (Springer-Verlag, New York, 1996).
[13] A. Yu. Ol’shanskii, ”Self-normalization of free subgroups in the free Burnside groups”, in: 179–187, 555, Groups, Rings, Lie and Hopf Algebras (St. John’s, NF, 2001), Math. Appl. (Kluwer Acad. Publ., Dordrecht, 2003).
[14] V. S. Atabekyan, ”Normalizers of free subgroups of free Burnside groups of odd order n 1003”, Fund. i Prikl. Matem., 15(1), 3–21 (2009). · Zbl 1226.20029
[15] V. S. Atabekyan, ”Normal subgroups in free Burnside groups of odd period”, Armen. J.Math., 1,2, 25–29 (2008). · Zbl 1281.20042
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