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On certain applications of the Khukhro-Makarenko theorem. (English) Zbl 1282.20031

Let \(F\) be a free group of countable rank with basis \(\{x_1,x_2,\ldots\}\). Then an outer commutator word of weight \(1\) is \(x_1\), and an outer commutator word \(\omega\) of weight \(t>1\) is a word of the form \(\omega(x_1,\ldots,x_t)=[u(x_1,\ldots,x_r),v(x_{r+1},\ldots,x_t)]\), where \(u,v\) are outer commutator words of weight \(r\) and \(t-r\), respectively. If \(\omega\) is an outer commutator word of weight \(t\), we denote by \(\mathfrak X_\omega\) the class of groups satisfying \(\omega(g_1,\ldots,g_t)=1\) for all \(g_1,\ldots,g_t\in G\), i.e. \(\omega (G)=1\). We denote by \(\mathfrak N_c\) the class of all nilpotent groups of nilpotency class \(c\).
The authors use the Khukhro-Makarenko theorem [E. I. Khukhro and N. Yu. Makarenko, J. Lond. Math. Soc., II. Ser. 75, No. 3, 635-646 (2007; Zbl 1132.20013); E. I. Khukhro, A. A. Klyachko, N. Yu. Makarenko and Yu. B. Melnikova, Bull. Lond. Math. Soc. 41, No. 5, 804-816 (2009; Zbl 1250.20023)] to obtain generalizations of some known results on groups, in which all proper subgroups satisfy certain conditions, in several cases the conditions in question being either “almost in the variety \(\mathfrak X_\omega\)” or “\(\mathfrak X_\omega\)-by-Chernikov”.
For example, the following theorem is proved. Let \(\omega\) be an outer commutator word of weight \(t>1\) and let \(G\) be a group all whose proper subgroups are \(\mathfrak X\)-by-finite but \(G\) itself is not \(\mathfrak X\)-by-finite. If \(G\) has no infinite simple images, then the following properties hold: (i) \(G\) has no proper subgroups of finite index and no simple images; (ii) \(N'\in\mathfrak X_\omega\) for every proper normal subgroup \(N\) of \(G\); (iii) \(G\) is not perfect, \(G'\in\mathfrak X_\omega\); in particular, \(G\) is solvable of derived length at most \(t\).
Another result of the paper is a theorem on barely transitive \(p\)-groups. Recall that a group \(G\) is ‘barely transitive’ if \(G\) has a subgroup \(H\) such that \(|G:H|\) is infinite, \(\bigcap_{g\in G}H^g=1\) and \(|K:K\cap H|\) is finite for every proper subgroup \(K\) of \(G\), where the subgroup \(H\) is called a ‘point stabiliser’. Let \(G\) be a locally finite barely transitive \(p\)-group with a point stabiliser \(H\) and let \(\omega\) be an outer commutator word of weight \(t\). It is proved that if \(H\in\mathfrak X_\omega\), then \(G'\neq G\) and \(G'\in\mathfrak X_\omega\).
Finally, the authors obtain some partial generalizations of the Khukhro-Makarenko theorem. In particular, it is proved that if a periodic group \(G\) has a normal subgroup \(N\in\mathfrak N_c\cap\mathfrak X_\omega\) such that \(G/N\) is a Chernikov group for some outer commutator word \(\omega\), then \(G\) contains a characteristic subgroup \(S\in\mathfrak N_c\cap\mathfrak X_\omega\) such that \(G/S\) is a Chernikov group.

MSC:

20F19 Generalizations of solvable and nilpotent groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E10 Quasivarieties and varieties of groups
20E07 Subgroup theorems; subgroup growth
20E34 General structure theorems for groups
20F50 Periodic groups; locally finite groups
20F12 Commutator calculus
20B07 General theory for infinite permutation groups
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