Dixon, M., Kurdachenko, L., Subbotin, I. (2018). On the relationships between the factors of the upper and lower central series in some non-periodic groups. International Journal of Group Theory, 7(1), 37-50. doi: 10.22108/ijgt.2017.21674

Martyn Dixon; Leonid Kurdachenko; Igor Subbotin. "On the relationships between the factors of the upper and lower central series in some non-periodic groups". International Journal of Group Theory, 7, 1, 2018, 37-50. doi: 10.22108/ijgt.2017.21674

Dixon, M., Kurdachenko, L., Subbotin, I. (2018). 'On the relationships between the factors of the upper and lower central series in some non-periodic groups', International Journal of Group Theory, 7(1), pp. 37-50. doi: 10.22108/ijgt.2017.21674

Dixon, M., Kurdachenko, L., Subbotin, I. On the relationships between the factors of the upper and lower central series in some non-periodic groups. International Journal of Group Theory, 2018; 7(1): 37-50. doi: 10.22108/ijgt.2017.21674

On the relationships between the factors of the upper and lower central series in some non-periodic groups

This paper deals with the mutual relationships between the factor group $G/\zeta(G)$ (respectively $G/\zeta_k(G)$) and $G'$ (respectively $\gamma_{k+1}(G)$ and $G^{\mathfrak{N}}$). It is proved that if $G/\zeta(G)$ (respectively $G/\zeta_k(G)$) has finite $0$-rank, then $G'$ (respectively $\gamma_{k+1}(G)$ and $G^{\mathfrak{N}}$) also have finite $0$-rank. Furthermore, bounds for the $0$-ranks of $G', \gamma_{k+1}(G)$ and $G^{\mathfrak{N}}$ are obtained.

[1] R. Baer, Representations of groups as quotient groups. II. minimal central chains of a group, Trans. Amer. Math. Soc., 58 (1945) 348–389.

[2] R. Baer, Endlichkeitskriterien für Kommutatorgruppen, Math. Ann., 124 (1952) 161–177.

[3] R. Baer and H. Heineken, Radical groups of finite abelian subgroup rank, Illinois J. Math., 16 (1972) 533–580.

[4] A. Ballester-Bolinches, S. Camp-Mora, L. A. Kurdachenko and J. Otal, Extension of a Schur theorem to groups with a central factor with a bounded section rank, J. Algebra, 393 (2013) 1–15.

[5] M. R. Dixon, L. A. Kurdachenko and J. Otal, On groups whose factor-group modulo the hypercentre has finite section p-rank, J. Algebra, 440 (2015) 489–503.

[6] M. R. Dixon, L. A. Kurdachenko and N. V. Polyakov, Locally generalized radical groups satisfying certain rank conditions, Ricerche di Matematica, 56 (2007) 43–59.

[7] M. R. Dixon, L. A. Kurdachenko and A. A. Pypka, The theorems of Schur and Baer: a survey, Int. J. Group Theory, 4 (2015) 21–32.

[8] M. De Falco, F. de Giovanni, C. Musella and Y. P. Sysak, On the upper central series of infinite groups, Proc. Amer. Math. Soc., 139 (2011) 385–389.

[9] V. M. Glushkov, On some questions of the theory of nilpotent and locally nilpotent groups without torsion, Mat. Sbornik N. S., 30 (1952) 79–104.

[10] L. Kaloujnine,Über gewisse Beziehungen zwischen einer Gruppe und ihren Automorphismen, Bericht über die Mathematiker-Tagung in Berlin, Januar, 1953, Deutscher Verlag der Wissenschaften, Berlin, 1953 164–172.

[11] M. I. Kargapolov, On solvable groups of finite rank, Algebra i Logika Sem., 1 (1962) 37–44.

[12] L. A. Kurdachenko and J. Otal, The rank of the factor-group modulo the hypercenter and the rank of the some hypocenter of a group, Cent. Eur. J. Math., 11 (2013) 1732–1741.

[13] , Groups with Chernikov factor-group by hypercentral, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 109 (2015) 569–579.

[14] L. A. Kurdachenko, J. Otal and A. A. Pypka, On some properties of central and generalized series of groups, Reports of the National Academy of Sciences of Ukraine, 1 (2015) 20–24.

[15] L. A. Kurdachenko, J. Otal and I. Ya. Subbotin, Artinian Modules over Group Rings, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.

[16] L. A. Kurdachenko and P. Shumyatsky, The ranks of central factor and commutator groups, Math. Proc. Cambridge Philos. Soc., 154 (2013) 63–69.

[17] A. I. Maltsev, Nilpotent torsion-free groups, Izvestiya Akad. Nauk. SSSR. Ser. Mat., 13 (1949) 201–212.

[18] B. H. Neumann, Groups with finite classes of conjugate elements, Proc. London Math. Soc. (3), 1 (1951) 178–187.

[19] D. J. S. Robinson, A new treatment of soluble groups with a finiteness condition on their abelian subgroups, Bull. London Math. Soc., 8 (1976) 113–129.

[20] A. Schlette, Artinian, almost abelian groups and their groups of automorphisms, Pacific J. Math., 29 (1969) 403–425.

[21] I. Schur,Über die Darstellungen der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math., 127 (1904) 20–50.

[22] D. I. Zaitsev, Hypercyclic extensions of abelian groups, Groups defined by properties of a system of subgroups (Russian), Akad. Nauk Ukrain. SSR, Inst. Mat. Kiev, 152 (1979) 16–37.