Positive laws in finite groups admitting a dihedral group of automorphisms. (Q2873312)

Let \(F\) denote the free group on \(X=\{x_1,x_2,\ldots\}\). A positive word in \(x_1,x_2,\ldots\) is a nontrivial element of \(F\) not involving the inverses of the \(x_i\). A positive law of a group \(G\) is a nontrivial identity of the form \(u\equiv v\) where \(u\), \(v\) are positive words in \(x_1,x_2,\ldots\) holding under every substitution \(X\to G\). The degree of such a law is the maximum of the lengths of \(u\), \(v\).NEWLINENEWLINE We quote the abstract: ``Suppose that a finite group \(G\) admits a dihedral group of automorphisms \(D=\langle\alpha,\beta\rangle\) generated by two involutions \(\alpha\) and \(\beta\) such that \(C_G(\alpha\beta)=1\). It is proved that if \(C_G(\alpha)\) and \(C_G(\beta)\) satisfy a positive law of degree \(k\), then \(G\) satisfies a positive law of degree bounded in terms of \(k\) and \(|D|\).''NEWLINENEWLINE Earlier \textit{P. Shumyatsky} [J. Algebra 375, 1-12 (2013; Zbl 1277.20023)] proved that if a finite group \(G\) admits a dihedral group of automorphisms \(D=\langle\alpha,\beta\rangle\) generated by two involutions \(\alpha\) and \(\beta\) such that \(C_G(\alpha\beta)=1\) and both \(C_G(\alpha)\) and \(C_G(\beta)\) are nilpotent of class \(c\), then \(G\) is nilpotent of class bounded in terms of \(c\). Another related result on centralizers of automorphisms satisfying a positive law can be found in [\textit{P. Shumyatsky}, J. Pure Appl. Algebra 215, No. 11, 2559-2566 (2011; Zbl 1235.20024)].