2-Engel implies class three for Lie rings

Statement

Suppose $L$ is a 2-Engel Lie ring. Then, $L$ is a nilpotent Lie ring of nilpotency class at most three.

Note that for the proof, we use the cyclic symmetry formulation of the 2-Engel condition: $[a,[b,c]]=[b,[c,a]]=[c,[a,b]]$ for all $a,b,c$ in the Lie ring.

Related facts

Similar facts

• Nilpotency class three is 3-local for Lie rings
• 2-Engel and 3-torsion-free implies class two for Lie rings
• 3-Engel and (2,5)-torsion-free implies class six for Lie rings
• 4-Engel and (2,3,5)-torsion-free implies nilpotent for Lie rings

Analogue in groups

• 2-Engel implies class three for groups

Applications

• n-local nilpotency class n implies nilpotency class n plus 1

Facts used

1. 2-Engel implies third member of lower central series is in 3-torsion for Lie rings: This say that $3[a,[b,c]]=0$ for all $a,b,c$ in the ring.

Proof

Direct proof

Given: Lie ring $L$ such that $[a,[b,c]]=[b,[c,a]]=[c,[a,b]]$ for all $a,b,c\in L$. This is the cyclic symmetry formulation of the 2-Engel condition.

To prove: $[w,[x,[y,z]]]=0$ for all $w,x,y,z\in L$.

Proof: We assume $w,x,y,z$ are fixed for the proof.

Step no.	Assertion/construction	Facts used	Given data used	Previous steps used	Explanation
1	$[[y,z],[w,x]]=[w,[x,[y,z]]]$		$L$ is 2-Engel (cyclic symmetry formulation)		Plug in $a=[y,z],b=w,c=x$ and use the cyclic symmetry formulation.
2	$[w,[x,[y,z]]]=[w,[y,[z,x]]]$		$L$ is 2-Engel (cyclic symmetry formulation)		Plug in $a=x,b=y,c=z$ and use the cyclic symmetry formulation.
3	$[w,[y,[z,x]]]=[y,[[z,x],w]]$		$L$ is 2-Engel (cyclic symmetry formulation)		Plug in $a=w,b=y,c=[z,x]$ and use the cyclic symmetry formulation.
4	$[y,[[z,x],w]]=-[y,[w,[z,x]]]$		$L$ is a Lie ring		Use skew symmetry between $[z,x]$ and $w$, and linearity in $y$.
5	$[y,[w,[z,x]]]=[y,[z,[x,w]]]$		$L$ is 2-Engel (cyclic symmetry formulation)		Plug in $a=w,b=z,c=x$ and use the cyclic symmetry formulation.
6	$[y,[z,[x,w]]]=-[y,[z,[w,x]]]$		$L$ is a Lie ring		Use skew symmetry between $x$ and $w$, and linearity in $y,z$.
7	$[y,[z,[w,x]]]=[[w,x],[y,z]]$		$L$ is 2-Engel (cyclic symmetry formulation)		Plug in $a=y,b=z,c=[w,x]$ and use the cyclic symmetry formulation.
8	$[[y,z],[w,x]]=[[w,x],[y,z]]$			steps (1)-(7)	Follows directly by combining these steps. Note that two steps involve negation and cancel each other out.
9	$[[y,z],[w,x]]=-[[w,x],[y,z]]$		$L$ is a Lie ring		Use skew symmetry between $[y,z]$ and $[w,x]$
10	$2[[y,z],[w,x]]=0$			Steps (8), (9)	Step-combination direct
11	$2[w,[x,[y,z]]]=0$			Steps (1), (10)
12	$3[w,[x,[y,z]]]=0$	Fact (1)	$L$ is 2-Engel		Note that Fact (1) shows that any bracket expression of length three is in the 3-torsion. The expression here has length four, but can be viewed as an expression of lenth three by setting $a=w,b=x,c=[y,z]$.
13	$[w,[x,[y,z]]]=0$			Steps (11), (12)	Subtract the equation of Step (11) from the equation of Step (12).

Proof via 3-local class

This is a somewhat shorter version of the above proof. The idea is to first use the 2-Engel condition to show that the 3-local class is at most three. Then, we use that nilpotency class three is 3-local for Lie rings.

References

Journal references

• Lie rings satisfying the Engel condition by P. J. Higgins, Proceedings of the Cambridge Philosophical Society, Volume 50, Page 8 - 15(Year 1954): ^{Weblink for official copyMore info}