COMMUTATOR IDENTITIES ON SYMMETRIC ELEMENTS OF GROUP ALGEBRAS

Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈ log 2 t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

THE TRANSFER OF A COMMUTATOR IDENTITY IN A NIL-GENERATED ALGEBRA

We show that if an associative algebra over a field of characteristic 0 is generated by its nilpotent elements and satisfies a multilinear Lie commutator identity then its adjoint group satisfies the corresponding multilinear group commutator identity.

Tarski's problem for varieties of groups with a commutator identity

It is proved that for a variety of groups in which the relatively free groups are solvable, the relatively free groups of distinct finite rank are not elementarily equivalent.

A commutator identity for basic p-groups

The consequences of the law (Ga(1), hellip, Ga(n)) = 1 in a basic p-group are examined. The principal tools are a combinatorial analysis of the lattice of n-tuples of positive integers and a theorem of the author about higher-commutator subgroups.Subject classification (Amer. Math. Soc. (MOS) 1970): 20 D 15.

Some symmetric varieties of groups

Let (n, σ, d) denote the variety of all groups defined by the left-normed commutator identity [x1, …, xn] = [x1σ, …, xnσ]d, where σ is a non-identity permutation of {1, …, n}, and d is an integer, possibly negative. It is shown that (n, σ, d) is nilpotent-by-nilpotent if σ ≠ (1, 2), abelian by nilpotent if n > 2, nσ ≠ n, and nilpotent of class at most n + 1 if {1, 2} ≠ {1σ, 2σ}. This improves on a result of E.B. Kikodze that (n, σ, 1) is locally soluble and if {1, 2} ≠ {1σ, 2σ} is locally nilpotent.