A generalization of the malnormal subgroups

1Kurdachenko, LA
2Semko, NN
3Subbotin, IYa.
1Oles Honchar Dnipropetrovsk National University
2State Tax Service National University of Ukraine, Irpin
3National University, Los Angeles, USA
Dopov. Nac. akad. nauk Ukr. 2019, 3:25-28
Section: Mathematics
Language: English

A subgroup H of a group G is called malonormal in G, if H Hx = ‹1› for every element x NG(H). These subgroups are generalizations of malnormal subgroups. Every malnormal subgroup is malonormal, and every selfnormalizing malonormal subgroup is malnormal. Furthermore, every normal subgroup is malonormal. In this paper we obtain a description of finite and certain infinite groups, whose subgroups are malonormal.

Keywords: Frobenius group, generalized radical groups, locally graded groups, malnormal subgroups, malonormal subgroups

1. Baumslag, B. (1968). Generalized free product whose twogenerator subgroups are free. J. London Math. Soc., 43, pp. 601-606. doi: https://doi.org/10.1112/jlms/s1-43.1.601
2. Gorenstein, D. (1980). Finite groups of prime power order. New York: Chelsea Publ. Co.
3. Flavell, P. (2000). A note on Frobenius groups. J. Algebra, 228, pp. 367-378. doi: https://doi.org/10.1006/jabr.2000.8269
4. De la Harpe, P. & Weber, C. (2014). Malnormal subgroups and Frobenius groups: basics and examples. Confluentes Math., 6, pp. 1, 65-76. doi: https://doi.org/10.5802/cml.13
5. Kirichenko, V. V., Kurdachenko, L. A. & Subbotin, I. Ya. (2011). Some related to pronormality subgroup families and the properties of a group. Algebra Discrete Math., 11, pp. 75-108.
6. Baer, R. (1933). Situation der Untergruppen und Struktur der Gruppe. S.B. Heidelberg Akad., 2, pp. 12-17.
7. Olshanskii, A. Yu. (1981). An infinite group with subgroups of prime orders. Math. USSRIzv., 16, pp. 279-289.
8. Chernikov, S. N. (1970). Infinite nonAbelian groups with normality condition for infinite nonabelian subgroups. Dokl. AN SSSR, 194, No. 6, pp. 1280-1283 (in Russian).