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Kurdachenko, Leonid A.; Smith, Howard

Groups with all subgroups either subnormal or self-normalizing. (English) Zbl 1078.20026

J. Pure Appl. Algebra 196, No. 2-3, 271-278 (2005).
Summary: Let \(G\) be a group in which every subgroup that is not self-normalizing is subnormal. It is shown that if \(G\) is locally finite then either \(G\) has a nilpotent subgroup of prime index or every subgroup of \(G\) is subnormal, while if \(G\) is not periodic then every subgroup of \(G\) is subnormal. If \(G\) is a periodic group with the given property then \(G\) need not be locally finite, but if \(G\) is also locally graded then it is locally finite.

Cited in 12 Documents

MSC:

20E15 Chains and lattices of subgroups, subnormal subgroups
20E07 Subgroup theorems; subgroup growth
20F50 Periodic groups; locally finite groups
20F18 Nilpotent groups
20E25 Local properties of groups

Keywords:

self-normalizing subgroups; subnormal subgroups; locally finite groups; nilpotent subgroups; locally graded groups
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\textit{L. A. Kurdachenko} and \textit{H. Smith}, J. Pure Appl. Algebra 196, No. 2--3, 271--278 (2005; Zbl 1078.20026)
Full Text: DOI

References:

[2] Casolo, C., Nilpotent subgroups of groups with all subgroups subnormal, B. Lond. Math. Soc., 35, 15-22 (2003) · Zbl 1025.20016
[3] Casolo, C., On the structure of groups with all subgroups subnormal, J. Group Theory, 5, 293-300 (2002) · Zbl 1002.20016
[4] deFalco, M.; Kurdachenko, L. A.; Subbotin, I. Ya., Groups with only abnormal and subnormal subgroups, Atti Semin. Mat. Fis. Univ. Modena, 47, 435-442 (1998) · Zbl 0918.20017
[5] Ebert, G.; Bauman, S., A note on subnormal and abnormal chains, J. Algebra, 36, 287-293 (1975) · Zbl 0314.20019
[7] Franciosi, S.; de Giovanni, F.; Kurdachenko, L. A., Groups with finite conjugacy classes of non-subnormal subgroups, Archiv Math., 70, 169-181 (1998) · Zbl 0948.20022
[8] Heineken, H.; Kurdachenko, L. A., Groups with subnormality for all subgroups that are not finitely generated, Annali Mat., 169, 203-232 (1995) · Zbl 0848.20023
[9] Kostrikin, A. I., Around Burnside (1986), Nauka: Nauka Moscow · Zbl 0624.17001
[10] Kurdachenko, L. A.; Smith, H., Groups with the maximal condition on non-subnormal subgroups, Bolletino U. Mat. Ital., 10B, 441-460 (1996) · Zbl 0851.20020
[11] Kurdachenko, L. A.; Smith, H., Groups with the weak minimal condition for non-subnormal subgroups, Annali Mat., 173, 299-312 (1997) · Zbl 0939.20040
[12] Kurdachenko, L. A.; Smith, H., Groups with the weak maximal condition for non-subnormal subgroups, Ricerche Mat, 47, 29-49 (1998) · Zbl 0928.20025
[13] Kurdachenko, L. A.; Smith, H., Groups in which all subgroups of infinite rank are subnormal, Glasgow Math. J., 46, 83-90 (2004) · Zbl 1059.20023
[14] Kurdachenko, L. A.; Smith, H., Groups with rank restrictions on non-subnormal subgroups, Turkish J. Math., 28, 165-176 (2004) · Zbl 1069.20017
[15] Lennox, J. C.; Stonehewer, S. E., Subnormal Subgroups of Groups (1987), Clarendon: Clarendon Oxford · Zbl 0606.20001
[16] Möhres, W., Torsionsgruppen, deren Untergruppen alle subnormal sind, Geom. Dedicata, 31, 237-244 (1989) · Zbl 0675.20022
[17] Möhres, W., Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind, Archiv Math., 54, 232-235 (1990) · Zbl 0663.20027
[18] Ol’shanskii, A. Yu., Geometry of Defining Relations in Groups (1989), Nauka: Nauka Moscow
[19] Phillips, R. E.; Wilson, J. S., On certain minimal conditions for infinite groups, J. Algebra, 51, 41-68 (1978) · Zbl 0374.20042
[21] Robinson, D. J.S., A Course in The Theory of Groups (1982), Springer: Springer Berlin · Zbl 0496.20038
[22] Roseblade, J. E., On groups in which every subgroup is subnormal, J. Algebra, 2, 402-412 (1965) · Zbl 0135.04901
[23] Shumyatsky, P., Locally finite groups with an automorphism whose centralizer is small, Topics in infinite groups, Quaderni di Matematica, 8, 278-296 (2002) · Zbl 1038.20014
[24] Smith, H., Residually nilpotent groups with all subgroups subnormal, J. Algebra, 244, 845-850 (2001) · Zbl 0988.20018
[25] Smith, H., Torsion-free groups with all subgroups subnormal, Archiv Math., 76, 1-6 (2001) · Zbl 0982.20018
[26] Smith, H., Torsion-free groups with all non-nilpotent subgroups subnormal, Topics in infinite groups, Quaderni di Matematica, 8, 297-308 (2001) · Zbl 1017.20017
[27] Smith, H., Groups with all non-nilpotent subgroups subnormal, Topics in infinite groups, Quaderni di Matematica, 8, 309-326 (2001) · Zbl 1017.20018
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