Locally finite p-groups with all subgroups either subnormal or nilpotent-by-Chernikov

Document Type: Research Paper

Authors

Abstract

‎We pursue further our‎ ‎investigation‎, ‎begun in [H. Smith‎, ‎Groups with all subgroups subnormal or‎ ‎nilpotent-by-Chernikov‎, Rend‎. ‎Sem‎. ‎Mat‎. ‎Univ‎. ‎Padova,126 (2011)‎, ‎245-253] and continued in [G. Cutolo and H. Smith‎, ‎Locally finite groups with all subgroups‎ ‎subnormal or nilpotent-by-Chernikov‎. ‎Centr‎. ‎Eur‎. ‎J‎. ‎Math., (to appear)] ‎‎‎of groups $G$ in which all subgroups are either subnormal or‎ ‎nilpotent-by-Chernikov‎. ‎Denoting by $\mathfrak{X}$ the class of all such‎ ‎groups‎, ‎our concern here is with locally finite p-groups in the class‎ ‎$\mathfrak{X}$‎, ‎where $p$ is a prime‎, ‎while an earlier article provided a‎ ‎reasonable classification of locally finite $\mathfrak{X}$ nb-groups in which‎ ‎all of the p-sections are nilpotent-by-Chernikov‎. ‎Our main result is that‎ ‎if $G$ is a Baer p-group in $\mathfrak{X}$ then $G$ is nilpotent-by-Chernikov‎.

Keywords

Main Subjects


A.O. Asar (2000). Locally nilpotent p-groups whose proper subgroups are hypercentral or nilpotent-by-Chernikov. J. London Math. Soc. (2). 61 (2), 412-422
C. Casolo (2002). On the structure of groups with all subgroups subnormal. J. Group Theory. 5 (3), 293-300
G. Cutolo and H. Smith Locally finite groups with all subgroups subnormal or nilpotent-by-Chernikov. Centr. Eur. J. Math..
E. Detomi (2003). Groups with many subnormal subgroups. J. Algebra. 264 (2), 385-396
M.R. Dixon, M.J. Evans and H. Smith (2000). Groups with all proper subgroups nilpotent-by-finite rank. Arch. Math. (Basel). 75 (2), 81-91
P. Hall (1958). Some sufficient conditions for a group to be nilpotent. Illinois J. Math.. 2, 787-801
W. Mohres (1989). Torsionsgruppen, deren Untergruppen alle subnormal sind. Geom. Dedicata. 31 (2), 237-244
W. Mohres (1990). Auflosbarkeit von Gruppen, deren Untergruppen alle subnormal sind. Arch. Math. (Basel). 54 (3), 232-235
F. Napolitani and E. Pegoraro (1997). On groups with nilpotent by Chernikov proper subgroups. Arch. Math. (Basel). 69 (2), 89-94
H. Smith (1983). Hypercentral groups with all subgroups subnormal. Bull. London Math. Soc.. 15 (3), 229-234
H. Smith (2001). Groups with all non-nilpotent subgroups subnormal. Topics in infinite groups, Quad. Mat. Dept. Math., Seconda Univ. Napoli, Caserta. 8, 309-326
H. Smith (2001). On non-nilpotent groups with all subgroups subnormal. Ricerche Mat.. 50 (2), 217-221
H. Smith (2011). Groups with all subgroups subnormal or nilpotent-by-Chernikov. Rend. Sem. Mat. Univ. Padova. 126, 245-253
H. Smith (2011). Groups that involve finitely many primes and have all subgroups subnormal. J. Algebra. 347 (1), 133-142