A generalization of Burnside's \(p\)-nilpotency criterion
In this report, the word ``group'' will be considered as a synonymous of ``finite group''. Let \(G\) be a group and let \(P\) be a Sylow \(p\)-subgroup of \(G\) for a prime~\(p\). Assume that \(\mathrm{N}_G(P)\) is \(p\)-nilpotent. A classical theorem of Burnside states that \(G\) is \(p\)-nilpotent if \(P\) is abelian. A theorem of \textit{P. Hall} [Proc.\ Lond.\ Math.\ Soc., II. Ser. 40, 468--480 (1935); 481--501 (1936; Zbl 0015.20201)] states that \(G\) is \(p\)-nilpotent if \(P\leq \mathrm{Z}_{p-1}(P)\), that is, the nilpotency class of \(P\) is less than~\(p\). Given a \(p\)-group \(P\) for a prime \(p\) and \(i\) an integer, we denote by \[ \Omega_i(P)=\langle x\in P\mid x^{p^i}=1\rangle \] and \(\Omega(P)\) will be \(\Omega_1(P)\) if \(p\) is odd and \(\Omega_2(P)\) if \(p=2\). A theorem of \textit{J. González-Sánchez} and \textit{T. S. Weigel} [Isr.\ J. Math. 181, 125--143 (2011; Zbl 1234.20024)] states that \(G\) is \(p\)-nilpotent if \(\Omega(P)\leq \mathrm{Z}_{p-1}(P)\). On the other hand, \textit{A. Ballester-Bolinches} and \textit{X. Guo} [J. Algebra 228, No.~2, 491--496 (2000; Zbl 0961.20016)] have shown that \(G\) is \(p\)-nilpotent if either \(\Omega(P\cap G')\leq \mathrm{Z}(P)\) or, when \(p=2\), \(\Omega_1(P\cap G')\leq \mathrm{Z}(P)\) and \(P\) is quaternion-free, where \(G'\) is the derived subgroup of~\(G\). Given two elements \(x\), \(y\in G\), the \(n\)-Engel word \(\mathrm{E}_n(x,y)\) is defined as \(\mathrm{E}_n(x,y)=[x,[x,[x,\dots, x,y]]\dots]\), with \(n\) copies of \(x\). If \(H\) and \(K\) are two subgroups of \(G\), we say that \(G\) satisfies the \(n\)-th Engel condition for \((H, K)\) if \(\mathrm{E}_n(h,k)=1\) for all \(h\in H\) and \(k\in K\). The main result of the paper under review extends all the above \(p\)-nilpotency criteria. They prove that, under the hypotheses in the first paragraph, \(G\) is \(p\)-nilpotent if either \(G\) satisfies the \((p-1)\)-th Engel condition for \((P, \Omega(P\cap \mathrm{O}^p(G)))\) or \(p=2\), \(\Omega_1(P)\leq \mathrm{Z}(P\cap \mathrm{O}^2(G))\) and \(P\) is quaternion-free. As a consequence, \(G\) is \(p\)-nilpotent if either \(\Omega(P\cap \mathrm{O}^p(G))\leq \mathrm{Z}_{p-1}(P)\) or \(p=2\), \(\Omega_1(P\cap \mathrm{O}^p(G))\leq \mathrm{Z}(P)\) and \(P\) is quaternion-free.
scientific article; zbMATH DE number 6718372
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