Let G be a non-abelian group and let Z (G) be the center of G. Associate a graph ΓG (called
non-commuting graph of G) with G as follows: Take G\Z (G) as the vertices of ΓG and join two …

We associate a graph Γ G to a nonlocally cyclic group G (called the noncyclic graph of G) as
follows: take G\Cyc (G) as vertex set, where Cyc (G)={x∊ G|〈 x, y〉 is cyclic for all y∊ G}, and …

We prove that the solvable radical of a finite group G coincides with the set of elements y
having the following property: for any x∈ G the subgroup of G generated by x and y is …

We present a survey of results on word equations in simple groups, as well as their
analogues and generalizations, which were obtained over the past decade using various …

This work gives a new approach to the construction of implicit operations. By considering
“higher-dimensional” spaces of implicit operations and implicit operators between them, the …

We characterise the class of finite solvable groups by two-variable identities in a way similar
to the characterisation of finite nilpotent groups by Engel identities. Let holds in G. We also …

A CHARACTERIZATION OF FINITE SOLUBLE GROUPS BY LAWS IN TWO VARIABLES Page
1 Bull. London Math. Soc. 37 (2005) 179–186 Cо2005 London Mathematical Society …

The essence of the non-commuting graph remind us to find a connection between this graph
and the commutativity degree as denoted by d (G). On the other hand, d (H, G) the relative …

We characterise the solvable groups in the class of finite groups by an inductively defined
sequence of two-variable identities. Our main theorem is the analogue of a classical …

In this paper we study the lattice of restricted subalgebras of a restricted Lie algebra. In
particular, we consider those algebras in which this lattice is dually atomistic, lower or upper …