Lie derived length and involutions in group algebras.
The main result of this article is the following. Theorem 1. Let \(p\) be an odd prime, \(G\) a group in which the set of \(p\)-elements forms a finite nonabelian subgroup and let \(FG\) be its group algebra over a field \(F\) of characteristic \(p\). Then NEWLINE\[NEWLINE\mathrm{dl}(FG_*^+)\geq\left\lceil\log_2(p+1)\right\rceilNEWLINE\]NEWLINE for every involution \(*\) which arises from \(G\).NEWLINENEWLINE The group algebra \(FG\) may be considered as a Lie algebra with the Lie operation defined by \([x,y]=xy-yx\) for all \(x,y\in FG\). For a subset \(S\subseteq FG\) we define the Lie derived series in the following way. Let \(\delta^{[0]}(S)\) be the vector space over \(F\) spanned by \(S\) and let \(\delta^{[n+1]}(S)\) be the vector space spanned by all Lie commutators \([x,y]\) with \(x,y\in\delta^{[n]}(S)\). The subset \(S\) is said to be Lie solvable if there exists \(n\) such that \(\delta^{[n]}(S)=0\), and the smallest such \(n\) is called the Lie derived length of \(S\) which will be denoted by \(\mathrm{dl}(S)\).NEWLINENEWLINE Let \(*\) be an involution on \(FG\). Denote \(FG_*^+\) and \(FG_*^-\) the set symmetric and skew-symmetric elements in \(FG\) under the involution \(*\), respectively, that is \(FG_*^+=\{x\in FG\mid x^*=x\}\) and \(FG_*^-=\{x\in FG\mid x^*=-x\}\). Evidently, they are vector spaces over \(F\). -- If \(x\) is a real number, then \(\left\lceil x\right\rceil\) is the least integer which is greater or equal to \(x\).NEWLINENEWLINE Theorem 1 and Proposition B of \textit{I. B. S. Passi, D. S. Passman} and \textit{S. K. Sehgal} [Can. J. Math. 25, 748-757 (1973; Zbl 0266.16011)] give immediately the following corollary.NEWLINENEWLINE Corollary 2. Let \(F\) be a field of characteristic \(p>2\) and \(G\) a group in which the set of \(p\)-elements forms a finite nonabelian subgroup with central derived subgroup of order \(p\). Then \(\mathrm{dl}(FG_*^+)=\left\lceil\log_2(p+1)\right\rceil\) for every involution \(*\) which arises from \(G\).
scientific article; zbMATH DE number 6094118