on the right n-engel group elements

On the Right and Left 4-engel Elements

In this paper we study left and right 4-Engel elements of a group. In particular, we prove that 〈a, a〉 is nilpotent of class at most 4, whenever a is any element and b are right 4-Engel elements or a are left 4-Engel elements and b is an arbitrary element of G. Furthermore we prove that for any prime p and any element a of finite p-power order in a group G such that a ∈ L4(G), a, if p = 2, and ...

un 2 00 4 Notes on Engel groups and Engel elements in groups . Some generalizations

Engel groups and Engel elements became popular in 50s. We consider in the paper the more general nil-groups and nil-elements in groups. All these notions are related to nilpotent groups and nilpo-tent radicals in groups. These notions generate problems which are parallel to Burnside problems for periodic groups. The first three theorems of the paper are devoted to nil-groups and Engel groups, w...

Group rings satisfying generalized Engel conditions

Let R be a commutative ring with unity of characteristic r≥0 and G be a locally finite group. For each x and y in the group ring RG define [x,y]=xy-yx and inductively via [x ,_( n+1)  y]=[[x ,_( n)  y]  , y]. In this paper we show that necessary and sufficient conditions for RG to satisfies [x^m(x,y)   ,_( n(x,y))  y]=0 is: 1) if r is a power of a prime p, then G is a locally nilpotent group an...

Left 3-engel Elements in Groups

In this paper we study left 3-Engel elements in groups. In particular, we prove that for any prime p and any left 3-Engel element x of finite p-power order in a group G, x is in the Baer radical of G. Also it is proved that 〈x, y〉 is nilpotent of class 4 for every two left 3-Engel elements in a group G.

on the probability of being a 2-engel group

‎let $g$ be a finite group and $d_2(g)$ denotes the probability ‎that $[x,y,y]=1$ for randomly chosen elements $x,y$ of $g$‎. ‎we ‎will obtain lower and upper bounds for $d_2(g)$ in the case where ‎the sets $e_g(x)={yin g:[y,x,x]=1}$ are subgroups of $g$ for ‎all $xin g$‎. ‎also the given examples illustrate that all the ‎bounds are sharp.

Right gw-Majorization on M_{n,m}