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Some sensitivity conditions for infinite groups. (English) Zbl 0774.20017
A subgroup $$K$$ of $$G$$ is called $$\chi$$-sensitive if all $$\chi$$-subgroups of $$K$$ are intersections of $$K$$ with $$\chi$$-subgroups of $$G$$. The authors consider sensitivity conditions for normal and for infinite subgroups. Here is a sample taken from their very interesting results:
(1) For a soluble periodic group $$G$$ all of its normal subgroups are pronormal-sensitive iff $$G$$ is a $$T$$-group and the subgroups of $$G$$ normalized by the 2-component of $$\gamma_ 3(G)$$ are pronormal (Proposition 2.5).
(2) If all infinite subgroups of $$G$$ are normal-sensitive, then all infinite subgroups of $$G$$ are $$T$$-groups; for soluble groups also the converse is true (Proposition 3.1).
(3) An infinite soluble group $$G$$ possesses only pronormal-sensitive infinite subgroups iff all subgroups of $$G$$ are pronormal or $$G$$ is a Prüfer-by-finite $$T$$-group (Theorem 3.5).
(4) An infinite soluble group $$G$$ possesses only maximal-sensitive infinite subgroups iff every infinite subgroup of $$G$$ is an intersection of maximal subgroups.
##### MSC:
 20E07 Subgroup theorems; subgroup growth 20F16 Solvable groups, supersolvable groups 20E28 Maximal subgroups 20F50 Periodic groups; locally finite groups