scholarly journals Left 3-Engel elements in groups of exponent 60

2018 ◽  
Vol 28 (04) ◽  
pp. 673-695 ◽  
Author(s):  
Gareth Tracey ◽  
Gunnar Traustason
Keyword(s):  
Nilpotent Radical ◽  
Engel Elements ◽  
Locally Nilpotent ◽  
Engel Element

Let [Formula: see text] be a group and let [Formula: see text] be a left [Formula: see text]-Engel element of order dividing [Formula: see text]. Suppose furthermore that [Formula: see text] has no elements of order [Formula: see text], [Formula: see text] and [Formula: see text]. We show that [Formula: see text] is then contained in the locally nilpotent radical of [Formula: see text]. In particular, all the left [Formula: see text]-Engel elements of a group of exponent [Formula: see text] are contained in the locally nilpotent radical.

A locally nilpotent radical of nonassociative rings

1971 ◽  
Vol 10 (4) ◽  
pp. 219-224 ◽  
Author(s):  
G. V. Dorofeev
Keyword(s):  
Nilpotent Radical ◽  
Locally Nilpotent

scholarly journals On a locally nilpotent radical Jacobson for special Lie algebras

2021 ◽  
Vol 22 (1) ◽  
pp. 234-272
Author(s):  
Olga Alexandrovna Pikhtilkova ◽  
Elena Vladimirovna Mescherina ◽  
Anna Nikolaevna Blagovisnava ◽  
Elena Vladislavovna Pronina ◽  
Olga Alekseevna Evseeva
Keyword(s):  
Lie Algebras ◽  
Nilpotent Radical ◽  
Locally Nilpotent

Classical Radicals of Invariants of Algebraic Skew Derivations

2022 ◽  
Vol 29 (01) ◽  
pp. 53-66
Author(s):  
Jeffrey Bergen ◽  
Piotr Grzeszczuk
Keyword(s):  
Hopf Algebra ◽  
Lie Algebra ◽  
Jacobson Radical ◽  
Prime Radical ◽  
Nilpotent Radical ◽  
Enveloping Algebra ◽  
Locally Nilpotent ◽  
Nil Radical

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

Locally nilpotent ideals of a Lie algebra

1967 ◽  
Vol 63 (2) ◽  
pp. 257-272 ◽  
Author(s):  
B. Hartley
Keyword(s):  
Lie Algebra ◽  
Characteristic Zero ◽  
Nilpotent Radical ◽  
Finite Dimensional ◽  
Locally Nilpotent

The purpose of this paper is to investigate the locally nilpotent radical of a Lie algebra L over a field of characteristic zero, its behaviour under derivations of L, and its behaviour with regard to finite-dimensional nilpotent subinvariant and ascendant subalgebras of L.

scholarly journals RIGHT 4-ENGEL ELEMENTS OF A GROUP

2010 ◽  
Vol 09 (05) ◽  
pp. 763-769 ◽  
Author(s):  
A. ABDOLLAHI ◽  
H. KHOSRAVI
Keyword(s):  
Nilpotent Groups ◽  
Normal Closure ◽  
Locally Nilpotent Groups ◽  
Engel Elements ◽  
Locally Nilpotent

We prove that the set of right 4-Engel elements of a group G is a subgroup for locally nilpotent groups G without elements of orders 2, 3 or 5; and in this case the normal closure ⟨x⟩G is nilpotent of class at most 7 for each right 4-Engel elements x of G.

scholarly journals Compact groups in which all elements have countable right Engel sinks

2020 ◽  
pp. 1-25
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Normal Subgroup ◽  
Compact Groups ◽  
Locally Nilpotent ◽  
Hausdorff Group ◽  
Engel Element

A right Engel sink of an element g of a group G is a set ${\mathscr R}(g)$ such that for every x ∈ G all sufficiently long commutators $[...[[g,x],x],\dots ,x]$ belong to ${\mathscr R}(g)$ . (Thus, g is a right Engel element precisely when we can choose ${\mathscr R}(g)=\{ 1\}$ .) It is proved that if every element of a compact (Hausdorff) group G has a countable right Engel sink, then G has a finite normal subgroup N such that G/N is locally nilpotent.

scholarly journals Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

2021 ◽  
Author(s):  
E. I. Khukhro ◽  
P. Shumyatsky
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Prime Order ◽  
Profinite Group ◽  
Nilpotent Subgroup ◽  
Profinite Groups ◽  
Locally Nilpotent ◽  
Engel Element

AbstractA right Engel sink of an element g of a group G is a set $${{\mathscr {R}}}(g)$$ R ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[g,x],x],\dots ,x]$$ [ . . . [ [ g , x ] , x ] , ⋯ , x ] belong to $${\mathscr {R}}(g)$$ R ( g ) . (Thus, g is a right Engel element precisely when we can choose $${{\mathscr {R}}}(g)=\{ 1\}$$ R ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set $${{\mathscr {E}}}(g)$$ E ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[x,g],g],\dots ,g]$$ [ . . . [ [ x , g ] , g ] , ⋯ , g ] belong to $${{\mathscr {E}}}(g)$$ E ( g ) . (Thus, g is a left Engel element precisely when we can choose $${\mathscr {E}}(g)=\{ 1\}$$ E ( g ) = { 1 } .) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$ φ of prime order such that every fixed point of $$\varphi $$ φ has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.

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