A combinatorial property of certain infinite groups. (English) Zbl 0803.20024
If \(k\geq 2\), let \(E_ k\) be the variety of groups satisfying the law \([x,{_ ky}]= 1\), i.e. of \(k\)-Engel groups. The authors continue their study of the wider class \(E^*_ k\): this consists of all groups \(G\) with the property that, for every pair of infinite subsets \(X\), \(Y\) of \(G\) there exists \(x \in X\) and \(y \in Y\) such that \([x,{_ ky}]= 1\). Here it is shown that an infinite \(E^*_ k\)-group is a \(k\)-Engel group if it is either locally soluble or locally finite. The second result depends on the classification of finite simple groups.
Reviewer: D.J.S.Robinson (Urbana)
MSC:
| 20F45 | Engel conditions |
| 20E10 | Quasivarieties and varieties of groups |
| 20E25 | Local properties of groups |
Keywords:
variety of groups; \(k\)-Engel groups; infinite subsets; infinite \(E^*_ k\)-group; locally soluble; locally finiteReferences:
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