## Locally Nilpotent p-Groups Whose Proper Subgroups Are Hypercentral-by-Chernikov

International Conference on Mathematics and Mathematics Education (ICMME-2018), Ordu University, Ordu, 27-29 June 2018, Ordu, Turkey, 27 - 29 June 2018, pp.72-73

• Publication Type: Conference Paper / Summary Text
• City: Ordu
• Country: Turkey
• Page Numbers: pp.72-73
• Gazi University Affiliated: Yes

#### Abstract

If is a group theoretical property or class of groups then a group G is a - group if G has the property or is a member of the class Let G be a group and be a property of groups. If every proper subgroup of G satisfies but G itsellf does not satisfy it, then G is called a minimal non- group (We denote the classes of minimal non- group by -group). In this work we study locally nilpotent minimal non- groups, where stands for hypercentral-by-Chernikov. It was shown in [1] that if N be a normal nilpotent subgroup and U be a nilpotent subnormal subgroup of any group then NU is nilpotent. In this study, a generalizations of this situation was given. Let N be a normal an N0 closed subgroup (see [2]) of G for a class N0- closed of groups and U be an N0-closed subnormal subgroup of G. Then UN is an N0-closed subgroup of G. In addition, the results for the nilpotent-byChernikovgroups of Asar [3] were also extended to hypercentral-by-Chernikov groups in this study. Thus, the following results were obtained. Let G be a Baer p-group and every proper subgroup is N0 closed -by-Chernikov for a class N0 closed . Then every proper subgroup of G which is generated by a subset of finite exponent is N0 closed . Also we show that if G is a Baer p-group and G has a normal hypercentral N subgroup such that G/N Chernikov. Then G/N is nilpotent. Key Words: -group, -groups, hypercentral-by-Chernikov grup, Bare group. REFERENCES [1] W. Möhres, Torsionsgruppen deren untergruppen alle subnormal Sind, Geom. Dedicata 31 (1989), 237-244. International Conference on Mathematics and Mathematics Education (ICMME-2018), Ordu University, Ordu, 27-29 June 2018 73 [2] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups -Vols. I, II (Springer, Berlin). [3] A.O. ASAR, Locally nilpotent p- groups whose proper subgroups are hypercentral or nilpotentby- Chernikov. J.London Math. Soc., 61 (2000), 412-422.