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Sequences of words characterizing finite solvable groups. (English) Zbl 1195.20016

Finite nilpotent groups can be characterised by means of the identity \([x_0,x_1,\dots,x_n]=1\) for some \(n\in\mathbb{N}\) and all \(x_i\in G\), \(0\leq i\leq n\), or, as M. Zorn has proved [in Bull. Am. Math. Soc. 42, 485-486 (1936; JFM 62.0088.10)], by means of the Engel identity \(e_n(x,y)=1\) for \(n\in\mathbb{N}\), where \(e_1(x,y)=[x,y]\) and \(e_n(x,y)=[e_{n-1}(x,y),y]\).
During many years it has been an open question whether solvability could be described by an identity involving only two variables. Recently two such sequences of words have been found: T. Bandman, G.-M. Greuel, F. Grunewald, B. Kunyavskiĭ, G. Pfister, E. Plotkin [Compos. Math. 142, No. 3, 734-764 (2006; Zbl 1112.20016)] gave the sequences \(u_n(x,y)\) defined by \(u_1(x,y)=x^{-2}y^{-1}x\) and \(u_n(x,y)=[xu_{n-1}(x,y)^{-1}x^{-1},yu_{n-1}(x,y)^{-1}y^{-1}]\), while J. N. Bray, J. S. Wilson, R. A. Wilson [Bull. Lond. Math. Soc. 37, No. 2, 179-186 (2005; Zbl 1075.20008)] found the sequence \(s_1(x,y)=x\), \(s_n(x,y)=[s_{n-1}(x,y)^{-y},s_{n-1}(x,y)]\).
In the paper under review, solvability of finite groups is characterised in terms of the following sequences:
(1) \(v_1=yx^2\), \(v_k=[v_{k-1}^{y^{-1}x^{-1}},v_{k-1}^{x^{-1}}]\);(2) \(v_1=yx^2\), \(v_k=[v_{k-1}^{y^{-1}x^{-1}},v_{k-1}^{yx}]\);
(3) \(v_1=xy\), \(v_k=[v_{k-1}^{yx^{-1}y^{-1}}, v_{k-1}^x]\);(4) \(v_1=xy\), \(v_k=[v_{k-1}^{yx^{-1}y^{-1}},v_{k-1}^{y^{-1}}]\);
(5) \(v_1=xy\), \(v_k=[v_{k-1}^{yx^{-1}y^{-1}},v_{k-1}^{xyx}]\);(6) \(v_1=xy\), \(v_k=[v_{k-1}^{yx^{-1}y^{-1}},v_{k-1}^{y^{-1}x^{-1}y^{-1}}]\).
These sequences characterise soluble groups in the sense that if there is an \(n\in\mathbb{N}\) such that \(v_n(x,y)=1\) for all \(x,y\in G\), then \(G\) is soluble.
A minimal counterexample to this theorem must be a simple group whose proper subgroups are soluble. These groups were classified by J. G. Thompson [Bull. Am. Math. Soc. 74, 383-437 (1968; Zbl 0159.30804)]. Some computations with SINGULAR or MAGMA, which are detailed in the PhD Thesis of the author, among other techniques of group theory, ring theory, and algebraic geometry, are used to show that there exist \(x,y\in G\) such that \(v_1(x,y)=v_2(x,y)\) and \(v_1(x,y)\neq 1\) (Lemma 2.1). This, combined with the techniques of Bandman et al., is enough to prove the main theorem.

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20E10 Quasivarieties and varieties of groups
20D06 Simple groups: alternating groups and groups of Lie type
20F12 Commutator calculus
20F14 Derived series, central series, and generalizations for groups
20F45 Engel conditions
14G05 Rational points
14G15 Finite ground fields in algebraic geometry

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References:

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