We compare the special rank of the factors of the upper central series and terms of the lower central series of a group. As a consequence we are able to show some generalizations of a theorem of Reinhold Baer.
This paper deals with one of the ways of studying infinite groups many of whose subgroups have a prescribed property, namely the consideration of minimal conditions. If P is a theoretical property of groups and subgroups, we show that a locally graded group P satisfies the minimal conditions for subgroups not having P if and only if either G is a Cernikov group or every subgroup of G satisfies P, for certain values of P concerning normality, nilpotency and related ideas.
In classifying certain infinite groups under minimal conditions it is needed to find non-simplicity criteria for the groups under consideration. We obtain some of such criteria as a consequence of the main result of the paper and the classification of finite simple groups.
This paper studies groups G whose all subgroups are either ascendant or self-normalizing. We characterize the structure of such G in case they are locally finite. If G is a hyperabelian group and has the property, we show that every subgroup of G is in fact ascendant provided G is locally nilpotent or non-periodic. We also restrict our study replacing ascendant subgroups by permutable subgroups, which of course are ascendant [Stonehewer S.E., Permutable subgroups of infinite groups, Math. Z., 1972,...
This article discusses the Leibniz algebras whose upper hypercenter has finite codimension. It is proved that such an algebra includes a finite dimensional ideal such that the factor-algebra is hypercentral. This result is an extension to the Leibniz algebra of the corresponding result obtained earlier for Lie algebras. It is also analogous to the corresponding results obtained for groups and modules.
2010 Mathematics Subject Classification: Primary 20N25; Secondary 08A72, 03E72.
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