Engel properties of group algebras. II
[For part I cf. Publ. Math. 49, No. 1-2, 183--192 (1996; Zbl 0862.16018).] The author characterizes modular group algebras with \(3\)-Engel groups of units, and modular group algebras of characteristic \(p\) with \(p\)-Engel and \(p-1\)-Engel groups of units. Let \(F\) be a field of prime characteristic \(p\), let \(G\) be a group with a nontrivial \(p\)-Sylow subgroup and denote by \(t_p(G)\) the set of \(p\)-elements of \(G\). Then: The group of units \(U(FG)\) is \(3\)-Engel if and only if one of the following conditions holds: (i) \(FG\) is Lie \(3\)-Engel; (ii) \(p=2\) and \(G\) is nilpotent of class \(2\) such that \(G'\) is elementary Abelian of order \(8\), and \(t_2(G)\) is of order \(8\) or \(16\) and central in \(G\); (iii) \(p=2\) and \(G\) is nilpotent of class \(2\) with \(G'\) elementary Abelian of order \(8\), and \(t_2(G)=\langle G',a\rangle\) is of order \(16\) such that \(|G:C_G(a)|=2\) and \(C_G'(a)=\langle a,G\rangle\); (iv) \(p=2\) and \(G\) is nilpotent of class \(2\) with \(G'=t_2(G)\) elementary Abelian of order \(16\); (v) \(p=2\) and \(G\) is nilpotent with \(G'=t_2(G)\) cyclic of order \(4\); (vi) \(p=2\) and \(G\) is nilpotent of class \(3\) with \(G'=t_2(G)\) elementary Abelian of order \(4\). Let \(G\) be a locally nilpotent nonabelian group. Then the group of units \(U(FG)\) is \(p\)-Engel if and only if \(G\) is nilpotent with commutator subgroup of order \(p\). If \(U(FG)\) is not \((p-2)\)-Engel, and \(U(FG)\) is \((p-1)\)-Engel if and only if \(G\) is nilpotent with \(G'=t_p(G)\) of order \(p\).
scientific article; zbMATH DE number 1308262