Vol 56, No 2 (2017)

It is proved that for any solvable subgroup G of an almost simple group S with simple socle isomorphic to An, n ≥ 5, there are elements x, y, z, t ∈ S such that G ∩ G^x ∩ G^y ∩ G^z ∩ G^t = 1.

We present a general criterion under which the equality var(A wr B) = var(A)var(B) holds for finite groups A and B. This generalizes some known results in this direction and continues our previous research [J. Alg., 313, No. 2, 455-458 (2007)] on varieties generated by wreath products of Abelian groups. The classification is based on the techniques developed by A. L. Shmel’kin, R. Burns, etc., who used critical groups, verbal wreath products, and Cross properties for studying critical groups in nilpotent-by-Abelian varieties.

We study universal theories of partially commutative Lie algebras whose defining graphs are cycles and trees. Within each of the two above-mentioned classes of partially commutative Lie algebras, necessary and sufficient conditions for the coincidence of universal theories are specified.