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

Analogue in groups

2-Engel implies class three for groups

Applications

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

Facts used

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 copy}^{More info}