Left 3-Engel elements in groups of exponent 5.
An element \(a\) of a group \(G\) is called left \(3\)-Engel if \([x,a,a,a]=1\) for all \(x\in G\) or equivalently \(\langle a,a^x\rangle\) is nilpotent of class at most \(2\) for all \(x\in G\). The main question about left \(3\)-Engel elements is the following Question. Does every left \(3\)-Engel element of a group lie in the Hirsch-Plotkin radical of the group? The author of the paper under review starts a systematic approach to the latter question. It is observed that a left \(3\)-Engel element \(a\) of a group \(G\) lies in the Hirsch-Plotkin radical of \(G\) if and only if \(K(a,n)=\langle a,a^{g_1},a^{g_2},\ldots,a^{g_n}\rangle\) is nilpotent for all \(n\) and all elements \(g_1,\ldots,g_n\). Let \(a\) be a left \(3\)-Engel element. Then the group \(K(a,n)\) has the property that \(\langle x,y^g\rangle\) is nilpotent of class at most \(2\) for all \(x,y\in\{a,a^{g_1},\ldots,a^{g_n}\}\) and all \(g\in G\). This is the motivation for the definition of sandwich groups: a group \(K\) is called a sandwich group if \(K=\langle X\rangle\) and \(\langle x,y^g\rangle\) is nilpotent of class at most \(2\) for all \(x,y\in X\) and \(g\in K\). The author proves that \(3\)-generated sandwich groups are nilpotent of class at most \(5\). The author first considers certain groups generated by three elements \(a,b,c\) such that \(\langle a,b\rangle\), \(\langle a,c\rangle\) and \(\langle b,c\rangle\) are nilpotent of classes \(r\), \(s\), \(t\), respectively. The latter groups are called of type \((r,s,t)\). The structure of the free group of type \((1,2,2)\) is completely determined by giving a power-commutator presentation. Next the structure of the free group of latter type with the extra condition that \(c\) is a left \(3\)-Engel element is determined. Then, the structure of the free group of type \((1,2,3)\) with extra conditions that \([b,c,c]=1\) and \(c\) is left \(3\)-Engel is determined. The proofs consist of tedious commutator calculus, however the author has tried to do the minimum works needed and I think he has succeeded to do so. One of the main results of the paper under review is the following: A left \(3\)-Engel element in a group of exponent \(5\) belongs to the Hirsch-Plotkin radical of the group. This is proved by the information obtained on the free 3-generator sandwich group and using similar techniques as in [\textit{M. Vaughan-Lee}, Proc. Lond. Math. Soc., III. Ser. 74, No. 2, 306-334 (1997; Zbl 0869.20027)]. Proving the nilpotency of the groups mentioned above has been done carefully by hand calculation. The next thing is to find ``good'' presentations of these nilpotent groups to play with them. The author finds these by hand calculation again; however they can be done by NQ package in {\mathsf GAP}.
scientific article; zbMATH DE number 6338798