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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.
20E07 Subgroup theorems; subgroup growth
20F16 Solvable groups, supersolvable groups
20E28 Maximal subgroups
20F50 Periodic groups; locally finite groups