Commutator identities on symmetric elements of group algebras. (Q2847689)

Let \(A\) be an associative algebra over a field \(K\) of odd characteristic \(p\) and let a group \(G\) be the direct product of a finite \(p\)-group \(P\neq 1\) and a Hamiltonian 2-group, i.e. a non-Abelian 2-group in which all subgroups are normal. Denote by \(S_*\) the set of all elements of \(S\subseteq A\) fixed by the involution \(*\) of \(A\). Then \(S_*\) is called the set of symmetric elements of \(S\) with respect to the involution \(*\). Denote by \(t(P)\) the nilpotency index of the augmentation ideal of the group algebra \(KP\).NEWLINENEWLINE The authors prove the following results.NEWLINENEWLINE Theorem 1. \dots \((KG)_*\) satisfies all Lie commutator identities of degree at least \(t(P)\), and \(U_*(KG)\) satisfies all group commutator identities of degree at least \(t(P)\). Furthermore, if \(P\) is powerful, then \((KG)_*\) satisfies no Lie commutator identity of degree less than \(t(P)\), and \(U_*(KG)\) satisfies no group commutator identity of degree less than \(t(P)\).NEWLINENEWLINE Corollary 1. \dots \((KG)_*\) is Lie nilpotent of index at most \(t(P)\), and \(U_*(KG)\) is nilpotent of class at most \(t(P)-1\). Furthermore, if \(P\) is powerful, then \(t_L((KG)_*)=cl(U_*(KG))+1= t(P)\).NEWLINENEWLINE Corollary 2. \dots \((KG)_*\) is Lie solvable of derived length at most \(\lceil\log_2 t(P)\rceil\), and \(U_*(KG)\) is solvable of derived length at most \(\lceil\log_2 t(P)\rceil\). Furthermore, if \(P\) is powerful, then \(dl_L((KG)_*)=dl(U_*(KG))=\lceil\log_2 t(P)\rceil\).