منابع مشابه
on the right n-engel group elements
in this paper we study right $n$-engel group elements. by modifying a group constructed by newman and nickel, we construct, for each integer $ngeq 5$, a 2-generator group $g =langle a, brangle$ with the property that $b$ is a right $n$-engel element but where $[b^k,_n a]$ is of infinite order when $knotin {0, 1}$.
متن کاملOn 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 ...
متن کاملun 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...
متن کاملGroup 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...
متن کاملEngel Graph Associated with a Group
Let G be a non-Engel group and let L(G) be the set of all left Engel elements of G. Associate with G a graph EG as follows: Take G\L(G) as vertices of EG and join two distinct vertices x and y whenever [x,k y] 6= 1 and [y,k x] 6= 1 for all positive integers k. We call EG, the Engel graph of G. In this paper we study the graph theoretical properties of EG.
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ژورنال
عنوان ژورنال: Journal of Algebra and Its Applications
سال: 2010
ISSN: 0219-4988,1793-6829
DOI: 10.1142/s0219498810004191