On non-periodic groups whose finitely generated subgroups are either permutable or pronormal

Kurdachenko, L. A. ; Subbotin, I. Ya. ; Ermolkevich, T. I.

Mathematica Bohemica, Tome 138 (2013) no. 1, pp. 61-74

Abstract (VO)

The current article considers some infinite groups whose finitely generated subgroups are either permutable or pronormal. A group $G$ is called a generalized radical, if $G$ has an ascending series whose factors are locally nilpotent or locally finite. The class of locally generalized radical groups is quite wide. For instance, it includes all locally finite, locally soluble, and almost locally soluble groups. The main result of this paper is the following\endgraf Theorem. Let $G$ be a locally generalized radical group whose finitely generated subgroups are either pronormal or permutable. If $G$ is non-periodic then every subgroup of $G$ is permutable.

Détail

DOI : 10.21136/MB.2013.143230

Classification : 20E07, 20E15, 20E25, 20E34, 20F14, 20F19, 20F22
Keywords: pronormal subgroup; permutable subgroup; finitely generated subgroup; abnormal subgroup

Bibliographie

[1] Baer, R.: Arrangement of subgroups and the structure of a group. Sitzungber. Heidelberger Akad. Wiss. 2 (1933), 12-17 German.

[2] Dedekind, R.: Groups with all normal subgroups. German Math. Ann. 48 (1897), 548-561.

[3] Dixon, M. R., Subbotin, I. Ya.: Groups with finiteness conditions on some subgroup systems: a contemporary stage. Algebra Discrete Math. No. 4 2009 (2009), 29-54. | MR | Zbl

[4] Ebert, G., Bauman, S.: A note of subnormal and abnormal chains. J. Algebra 36 (1975), 287-293. | DOI | MR

[5] Falco, M. De, Kurdachenko, L. A., Subbotin, I. Ya.: Groups with only abnormal and subnormal subgroups. Atti Sem. Mat. Fis. Univ. Modena 47 (1998), 435-442. | MR | Zbl

[6] Gruenberg, K. W.: The Engel elements of soluble groups. Illinois J. Math. 3 (1959), 151-168. | DOI | MR

[7] Fattahi, A.: Groups with only normal and abnormal subgroups. J. Algebra 28 (1974), 15-19. | DOI | MR | Zbl

[8] Hall, P.: Some sufficient conditions for a group to be nilpotent. Illinois J. Math. 2 (1958), 787-801. | DOI | MR | Zbl

[9] Kurdachenko, L. A., Otal, J., Subbotin, I. Ya.: Artinian Modules over Group Rings. Birkhaüser, Basel (2007). | MR | Zbl

[10] Kurdachenko, L. A., Smith, H.: Groups with all subgroups either subnormal or self-normalizing. J. Pure Appl. Algebra 196 (2005), 271-278. | DOI | MR | Zbl

[11] Kurdachenko, L. A., Subbotin, I. Ya., Chupordya, V. A.: On some near to nilpotent groups. Fundam. Appl. Math. 14 (2008), 121-134. | MR

[12] Kurdachenko, L. A., Subbotin, I. Ya., Ermolkevich, T. I.: Groups whose finitely generated subgroups are either permutable or pronormal. Asian-European J. Math. 4 (2011), 459-473. | DOI | MR | Zbl

[13] Kuzennyi, N. F., Subbotin, I. Ya.: New characterization of locally nilpotent $ \overline{IH}$-groups. Russian Ukrain. Mat. J. 40 (1988), 322-326. | MR

[14] Kuzennyi, N. F., Subbotin, I. Ya.: Locally soluble groups in which all infinite subgroups are pronormal. Russian Izv. Vyssh. Ucheb. Zaved., Mat. 11 (1988), 77-79. | MR

[15] Legovini, P.: Finite groups whose subgroups are either subnormal or pronormal. Italian Rend. Semin. Mat. Univ. Padova 58 (1977), 129-147. | MR

[16] Legovini, P.: Finite groups whose subgroups are either subnormal or pronormal. II. Italian Rend. Semin. Mat. Univ. Padova 65 (1981), 47-51. | MR | Zbl

[17] Miller, G. A., Moreno, H. C.: Non-abelian groups in which every subgroup is abelian. Trans. Amer. Math. Soc. 4 (1903), 389-404. | DOI | MR

[18] Olshanskii, A. Yu.: Geometry of Defining Relations in Groups. Kluwer Acad. Publ., Dordrecht (1991). | MR

[19] Peng, T. A.: Finite groups with pronormal subgroups. Proc. Amer. Math. Soc. 20 (1969), 232-234. | DOI | MR

[20] Plotkin, B. I.: Radical groups. Russian Mat. Sbornik 37 (1955), 507-526. | MR | Zbl

[21] Rose, J. S.: Nilpotent subgroups of finite soluble groups. Math. Z. 106 (1968), 97-112. | DOI | MR | Zbl

[22] Schmidt, O. Yu.: Groups whose all subgroups are special. Russian Mat. Sbornik 31 (1925), 366-372.

[23] Schmidt, R.: Subgroups Lattices of Groups. Walter de Gruyter, Berlin (1994).

[24] Shemetkov, L. A.: Formations of Finite Groups. Russian Nauka, Moskva (1978). | MR | Zbl

[25] Stonehewer, S. E.: Permutable subgroups of infinite groups. Math. Z. 126 (1972), 1-16. | DOI | MR | Zbl

[26] Zacher, G.: Finite soluble groups in which composition subgroups are quasi-normal. Italian Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 37 (1964), 150-154.