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On the quasicentralizer condition. (English) Zbl 0984.20023

The quasicentralizer \(Q_G(A)\) of the subgroup \(A\) in the group \(G\) is the set of all the elements in \(G\) normalizing every subgroup of \(A\) (see, for example, the textbook of W. Scott [Group theory (1964; Zbl 0126.04504)]). The structure of groups with quasicentralizer condition for normal subgroups (for short, \(I(N)\)-groups), that is, of groups in which for every proper normal subgroup \(N\) there exists an element \(x\in N\), normalizing every subgroup of \(N\), has been described by I. Ya. Subbotin [Studies in group theory, Work Collect., Kiev 1976, 139-161 (1976; Zbl 0416.20013), ibid. 1978, 94-117 (1978; Zbl 0444.20023), ibid. 1979, 106-126 (1979; Zbl 0427.20027), Izv. Vyssh. Uchebn. Zaved., Mat. 1983, No. 9(256), 56-64 (1983; Zbl 0531.20019) and ibid. 1984, No. 11(270), 40-45 (1984; Zbl 0572.20017)]. The notion of \(I(N)\)-groups can be generalized if one considers not all the normal subgroups \(N\) in \(G\) but a specific kind of normal subgroups. For example, I. Ya. Subbotin and N. F. Kuzennyj [Work Collect., Kiev 1986, 101-107 (1986; Zbl 0593.20033) and Izv. Vyssh. Uchebn. Zaved., Mat. 1987, No. 10(305), 68-70 (1987; Zbl 0634.20010)] studied the finite groups with quasicentralizer condition for all the normal non-Abelian subgroups.
In the article under review the authors consider the groups with quasicentralizer condition for normal subgroups \(N\) which define factor groups \(G/N\) belonging to a certain kind of varieties, namely, to \(A\)-varieties. A variety \(\mathcal X\) is said to be an \(A\)-variety if (i) \(\mathcal X\) contains the variety \(\mathcal A\) of all Abelian groups, and (ii) if \(\mathcal X\) does not contain any simple non-Abelian group (the variety of all Abelian groups, the variety of all \(n\)-Engel groups, the variety of all soluble groups of length at most \(k\) are examples of \(A\)-varieties). An \(I(A)\)-group is a group with quasicentralizer condition for all the proper normal subgroups of the type mentioned.
The main result of the article is the following theorem describing \(I(A)\)-groups: an arbitrary group \(G\) is an \(I(A)\)-group if and only if: (i) \(G\) is an \(I(N)\)-group; (ii) \(G\) is a periodic group and \(G=AB\), where \([A,A]=A=G''\not=1\), \(B=Q_G(G')\), \([A,B]=1\), \(B\) is a soluble \(I(N)\)-group, \(\pi(A)\cap\pi(B)\) is a subset of the set \(\{2\}\), and if \(2\in\pi(B')\) then \(A\) does not have an element of order \(4\); (iii) \(G\) is a non-periodic group with \(G=CD\) where \(C=C'=G'\not=1\), \(D=\zeta G\).

MSC:

20F16 Solvable groups, supersolvable groups
20E10 Quasivarieties and varieties of groups
20E07 Subgroup theorems; subgroup growth
20E15 Chains and lattices of subgroups, subnormal subgroups
20E22 Extensions, wreath products, and other compositions of groups
20F50 Periodic groups; locally finite groups
20F14 Derived series, central series, and generalizations for groups
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