Engel group: Difference between revisions

Latest revision as of 15:07, 3 December 2024

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group $G$ is termed an Engel group or nil group or nilgroup, if, given any two elements $x, y \in G$ , there exists a $n$ such that the iterated commutator:

$[[\dots [x, y], y], y], \dots], y] = e$

where $e$ denotes the identity element, $[x, y] = x y x^{- 1} y^{- 1}$ denotes the commutator of $x$ and $y$ , and $y$ occurs $n$ times.

If there exists a $n$ that works for all pairs of elements of $G$ , then we say that $G$ is a $n$ -Engel group. A $n$ -Engel group, for some $n$ , is termed a bounded Engel group. Note that sometimes the term Engel group is used for bounded Engel group.

Note if we instead define the commutator as $[x, y] = x^{- 1} y^{- 1} x y$ we get an equivalent definition.

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
locally nilpotent group	every finitely generated subgroup is nilpotent	\|FULL LIST, MORE INFO
nilpotent group		\|FULL LIST, MORE INFO
bounded Engel group		\|FULL LIST, MORE INFO
2-locally nilpotent group	subgroup generated by two elements is always nilpotent	\|FULL LIST, MORE INFO

Examples and counterexamples

Finite groups

All the nilpotent groups, which are equivalent to locally nilpotent groups for finite groups, are Engel groups.

The smallest non-Engel finite group is symmetric group:S3. To see that, consider $x = (1 3)$ , $y = (2 3)$ . Define $x_{0} = x$ , $x_{i} = [x_{i - 1}, y]$ for $i \geq 1$ and you will see that none of $x_{0}, x_{1}, x_{2}, \dots$ are the identity permutation. (The values of $x_{0}, x_{1}, x_{2}, \dots$ depend on which definition of commutator you use, and which convention you take on the order of composition of permutations)

@@ Line 3: / Line 3: @@
 ==Definition==
-A [[group]] <math>G</math> is termed an '''Engel group''' or '''nil group''' or '''nilgroup''', if, given any two elements <math>x,y \in G</math>, there exists a <math>n</math> such that the iterated commutator:
+A [[group]] <math>G</math> is termed an '''Engel group''' or '''nil group''' or '''nilgroup''', if, given any two elements <math>x,y \in G</math>, there exists a <math>n</math> such that the iterated [[commutator]]:
 <math>[[ \dots [x,y],y],y],\dots],y] = e</math>
-where <math>e</math> denotes the identity element, and <math>y</math> occurs <math>n</math> times.
+where <math>e</math> denotes the identity element, <math>[x,y] = xyx^{-1}y^{-1}</math> denotes the commutator of <math>x</math> and <math>y</math>, and <math>y</math> occurs <math>n</math> times.
 If there exists a <math>n</math> that works for all pairs of elements of <math>G</math>, then we say that <math>G</math> is a <math>n</math>-Engel group. A <math>n</math>-Engel group, for some <math>n</math>, is termed a [[bounded Engel group]]. Note that sometimes the term ''Engel group'' is used for bounded Engel group.
+Note if we instead define the commutator as <math>[x,y] = x^{-1}y^{-1}xy</math> we get an equivalent definition.
 ==Relation with other properties==
@@ Line 18: / Line 20: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Weaker than::Locally nilpotent group]] || every [[finitely generated group|finitely generated]] subgroup is [[nilpotent group|nilpotent]] || || || {{intermediate notions short|locally nilpotent group|Engel group}}
+| [[Weaker than::locally nilpotent group]] || every [[finitely generated group|finitely generated]] subgroup is [[nilpotent group|nilpotent]] || || || {{intermediate notions short|Engel group|locally nilpotent group}}
 |-
-| [[Weaker than::Nilpotent group]] || || || || {{intermediate notions short|nilpotent group|Engel group}}
+| [[Weaker than::nilpotent group]] || || || || {{intermediate notions short|Engel group|nilpotent group}}
 |-
-| [[Weaker than::Bounded Engel group]] || || || || {{intermediate notions short|nilpotent group|bounded Engel group}}
+| [[Weaker than::bounded Engel group]] || || || || {{intermediate notions short|Engel group|bounded Engel group}}
+|-
+| [[Weaker than::2-locally nilpotent group]] || subgroup generated by two elements is always nilpotent || || || {{intermediate notions short|Engel group|2-locally nilpotent group}}
 |}
+==Examples and counterexamples==
+===Finite groups===
+* All the [[nilpotent group]]s, which are equivalent to [[locally nilpotent group]]s for finite groups, are Engel groups.
+* The smallest non-Engel finite group is [[symmetric group:S3]]. To see that, consider <math>x = (1 \, 3)</math>, <math>y = (2 \, 3)</math>. Define <math>x_0 = x</math>, <math>x_i = [x_{i-1}, y]</math> for <math>i \geq 1</math> and you will see that none of <math>x_0, x_1, x_2, \dots</math> are the identity permutation. (The values of <math>x_0, x_1, x_2, \dots</math> depend on which definition of commutator you use, and which convention you take on the order of composition of permutations)