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
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
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[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.