Details

Mattia Brescia ^[a] and Francesco de Giovanni ^[b]

Groups satisfying the double chain condition on non-pronormal subgroups

Pages: 353-366
Received: 27 July 2017
Accepted: 7 November 2017
Mathematics Subject Classification (2010): 20E15
Keywords: Double chain condition, pronormal subgroup, \(T\)-group
Author address:
[a],[b]: Università di Napoli Federico II, Dipartimento di Matematica e Applicazioni, via Cintia, Napoli, 80126, Italy

Abstract: If \(\theta\) is a subgroup property, a group \(G\) is said to satisfy the double chain condition on \(\theta\)-subgroups if it admits no infinite double sequences

consisting of \(\theta\)-subgroups. The structure of generalized soluble groups satisfying the double chain condition on non-pronormal subgroups is investigated.

References

[1]: M. Brescia and F. de Giovanni, Groups satisfying the double chain condition on subnormal subgroups, Ric. Mat. 65 (2016), 255-261. MR3513912
[2]: S. N. Černikov, Infinite nonabelian groups with a minimal condition for non-normal subgroups, Math. Notes 6 (1969), 465-468. DOI: 10.1007/BF01450247
[3]: M. Contessa, On rings and modules with \(DICC\), J. Algebra 101 (1986), 489-496. MR0847173
[4]: C. D. H. Cooper, Power automorphisms of groups, Math. Z. 107 (1968), 335-356. MR0236253
[5]: G. Cutolo, On groups satisfying the maximal condition on nonnormal subgroups, Riv. Mat. Pura Appl. 9 (1991), 49-59. MR1133590
[6]: F. De Mari and F. de Giovanni, Double chain conditions for infinite groups, Ricerche Mat. 54 (2005), 59-70. MR2290204
[7]: W. Gaschütz, Gruppen, in denen das Normalteilersein transitiv ist., J. Reine Angew. Math. 198 (1957), 87-92. MR0091277
[8]: F. de Giovanni and G. Vincenzi, Groups satisfying the minimal condition on non-pronormal subgroups, Boll. Un. Mat. Ital. A (7) 9 (1995), 185-194. MR1324619
[9]: F. de Giovanni and G. Vincenzi, Pronormality in infinite groups, Math. Proc. R. Ir. Acad. 100A (2000), 189-203. MR1883103
[10]: F. de Giovanni and G. Vincenzi, Some topics in the theory of pronormal subgroups of groups, in: "Topics in infinite groups", Quad. Mat., 8, Seconda Univ. Napoli, Caserta, 2001, 175-202. MR1949564
[11]: A. V. Jategaonkar, Integral group rings of polycyclic-by-finite groups, J. Pure Appl. Algebra 4 (1974), 337-343. MR0344345
[12]: N. F. Kuzennyi and I. Ya. Subbotin, Groups in which all subgroups are pronormal, Ukrain. Math. J. 39 (1987), 251-254. DOI: 10.1007/BF01057228
[13]: N. F. Kuzennyi and I. Ya. Subbotin, Locally soluble groups in which all infinite subgroups are pronormal, Soviet Math. (Iz. VUZ) 32 (1988), 126-131. MR0983287 zbMATH
[14]: J. C. Lennox and D. J. S. Robinson, The theory of infinite soluble groups, The Clarendon Press, Oxford, 2004. MR2093872
[15]: G. Navarro, Pronormal subgroups and zeros of characters, Proc. Amer. Math. Soc. 142 (2014), 3003-3005. MR3223355
[16]: T. A. Peng, Finite groups with pro-normal subgroups, Proc. Amer. Math. Soc. 20 (1969), 232-234. MR0232850
[17]: D. J. S. Robinson, Groups in which normality is a transitive relation, Proc. Cambridge Philos. Soc. 60 (1964), 21-38. MR0159885
[18]: D. J. S. Robinson, Finiteness conditions and generalized soluble groups, Part 1 & Part 2, Springer-Verlag, Berlin 1972. MR0332989 MR0332990
[19]: D. J. S. Robinson, A course in the theory of groups, 2nd edition, Grad. Texts in Math., 80, Springer-Verlag, New York, 1996. MR1357169
[20]: J. S. Rose, Finite soluble groups with pronormal system normalizers, Proc. London Math. Soc. (3) 17 (1967), 447-469. MR0212092
[21]: T. S. Shores, A chain condition for groups, Rocky Mountain J. Math. 3 (1973), 83-89. MR0347984
[22]: G. Vincenzi, Groups satisfying the maximal condition on non-pronormal subgroups, Algebra Colloq. 5 (1998), 121-134. MR1682978
[23]: G. J. Wood, On pronormal subgroups of finite soluble groups, Arch. Math. (Basel) 25 (1974), 578-585. MR0379660
[24]: D. I. Zaicev, Theory of minimax groups, Ukrainian Math. J. 23 (1971), 536-542. DOI: 10.1007/BF01091652
[25]: D. I. Zaicev, On solvable subgroups of locally solvable groups, Soviet Math. Dokl. 15 (1974), 342-345. MR0338181