For each element u in a finite group G define map θ : → by ( g ) = [ − , ] and set Θ { ∈ | n for some > 0 } . Then induces permutation of ; let β be the number orbits apart from {1}. Building on work J.N. Bray, R.A. Wilson second author, we show that index soluble radical is bounded terms values 2-elements

The calculus of classes and closure operations has proved to be a useful tool in group theory led deep the study finite soluble groups. More recently, parallel theories have started developed various varieties algebras, such as Lie, Leibniz Malcev algebras. This paper seeks investigate extent which these later can generalized variety all non-associative

Let nand k be positive integers. We say that a group G satisfies the condition £(n) (respectively, £dn)) if and only if any set with n + 1 elements of G contains two distinct elements x, y such that [x,t y] = 1 for some positive integer t = t(x, y) (respectively, [X,k y] = 1). Here we study certain groups satisfying these conditions. We prove that if G is a finite group satisfying the condition...

We describe under various conditions abelian subgroups of the automorphism group Aut(Tn) of the regular n-ary tree Tn, which are normalized by the n-ary adding machine τ = (e, . . . , e, τ)στ where στ is the n-cycle (0, 1, . . . , n− 1). As an application, for n = p a prime number, and for n = 4, we prove that every soluble subgroup of Aut(Tn), containing τ is an extension of a torsion-free met...