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On certain pairs of non-Engel elements in finite groups. (English) Zbl 1282.20029

Let \(G\) be a group, let \(g,h\in G\) and \(n>0\). The 2-tuple \((g,h)\) is called an \(n\)-Engel pair, if \(g=[g,{_nh}]\) and \(h=[h,{_ng}]\). Groups \(G\) are studied that can be generated by an \(n\)-Engel pair. These are quotients of \[ G(n,n)=\langle x,y\mid x=[x,{_ny}],\;y=[y,{_nx}]\rangle. \] In the paper under review, groups \(G_n\) with an additional relation are considered: \[ G_n=\langle x,y\mid x=[x,{_ny}],\;y=[y,{_nx}],\;yxy=xyx\rangle. \] Thus \(G_n\) is a quotient of \(G(n,n)\). By elegant and tricky computations, it is shown that \(G_n=1\) if \(n\not\equiv 0\bmod 5\), and all the remaining groups \(G_n\) are isomorphic: \(G_{5\ell}\cong G_5\cong\text{SL}(2,5)\) for all positive integers \(\ell\).
Remark: The paper sheds some light on the reviewer’s conjecture \(G(5,5)\cong\text{SL}(2,5)\) (see also Problem 11.18 in The Kourovka notebook [V. D. Mazurov (ed.) and E. I. Khukhro (ed.), The Kourovka notebook. Unsolved problems in group theory. 14th ed., Novosibirsk: Institut Matematiki SO RAN (1999; Zbl 0943.20003)]). Also, a closer study of the relation (4) in the paper perhaps can be used to tackle (Problem 11.17b) of [loc. cit.].
Reviewer: Rolf Brandl (Hof)

MSC:

20F05 Generators, relations, and presentations of groups
20F45 Engel conditions
20F12 Commutator calculus
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