# Proving the 5-Engel identity in the
2-generator group of exponent four

Colin Ramsay  School of Electrical Engineering and Computer Science, The
University of Queensland, Queensland 4072, Australia<EMAIL_ADDRESS>
###### Abstract.

It is known that the fifth Engel word $E_{5}$ is trivial in the 2-generator
group of exponent four $B(2,4)$, and so can be written as a product of fourth
powers. Explicit products of 250 and 28 powers are known, using fourth powers
of words up to lengths four and ten respectively. Using a reduction technique
based on the recursive enumerability of the set of trivial words in a finite
presentation we were able to rewrite $E_{5}$ as a product of 26 fourth powers
of words up to length five.

###### Key words and phrases:

Fifth Engel word, Group of exponent four, Product of fourth powers

###### 2010 Mathematics Subject Classification:

Primary: 20F10; Secondary: 20F45, 20-08

##  1\. Introduction

If $x$ and $y$ are elements of a group $G$, then the commutator
$[x,y]=x^{-1}y^{-1}xy$. The commutator is also known as the first Engel word
$E_{1}$. The $n$th Engel word $E_{n}$, $n>1$, is defined as $[E_{n-1},y]$. $G$
it is said to satisfy the $n$-Engel identity if $E_{n}$ is trivial in $G$.

Burnside [1] introduced what are now called the free Burnside groups of
exponent $n$ on $d$ generators $B(d,n)$. The group $B(2,4)$ is finite, with
order $2^{12}$, and satisfies the 5-Engel identity. So in a free group, and
hence any group, $E_{5}$ can be expressed as a product of fourth powers. An
explicit expression for $E_{5}$ was first given in [2], and contained 250
fourth powers. A shorter expression, with 28 fourth powers, was given in [3].

Given a finite presentation $P=\langle S|R\rangle$ for a group $G$ it is, in
general, an unsolvable problem to decide whether or not a given element $g\in
G$ is trivial. As $S$ and $R$ are finite, the set of trivial words in $G$ is
recursively enumerable. (This is also true if $S$ is finite and $R$ itself is
recursively enumerable, but we consider only finite presentations here.)

In Section 2 we describe a simple method of generating trivial words in $P$,
and prove that this enumerates the set of all trivial words. This engenders a
reduction technique for generating proofs that a particular word in the free
group generated by $S$ is trivial in $G$. Using this, we were able to generate
a new shortest proof that $E_{5}$ is trivial in $B(2,4)$. Our proof uses 26
fourth powers (instead of 28), and utilises words up to length five (instead
of ten). We discuss this proof, and the proofs given in [2, 3], in Section 3.

##  2\. Generating Proofs

Let $F$ be the free group with generating set $S$, and let $R$ be a set of
words in $F$. Let $R^{\prime}$ be the normal closure of the subgroup of $F$
generated by all words $r\in R$. The group $G$ presented by $\langle
S|R\rangle$ is the quotient group $F/R^{\prime}$. The elements of $G$ are the
cosets of $R^{\prime}$, with the words in $R^{\prime}$ being the words in $F$
which are trivial in $G$. So a non-empty freely reduced word $w$ in $F$ is
trivial in $G$ if and only if it can be written as a product of conjugates of
relators

(1) $w=\prod^{N}_{i=1}\,u_{i}^{-1}\,r_{i}^{\pm 1}\,u_{i},$

where $r_{i}\in R$ and the $u_{i}$ are words in $F$.

The decomposition of $w$ on the right hand side of (1) is a proof that $w$ is
trivial in $G$, and is called a _proof word_. It freely reduces to $w$ and if
the $r_{i}$ are excised (being trivial in $G$) the remainder freely reduces to
the empty word $\varepsilon$, and thus $w$ is trivial in $G$.

The proof word in (1) is in the form of individually conjugated relators.
There is typically significant free cancellation in the words
$u_{i-1}u_{i}^{-1}$, $2\leqslant i\leqslant N$, and we normally work with
proof words where these words, and $u_{1}^{-1}$ and $u_{N}$, are freely
reduced. We call these _reduced proof words_ , and note that to enumerate all
the trivial words in $G$ it suffices to enumerate only the reduced proof
words.

One way to construct reduced proof words is to start with some relator $r\in
R$ and then perform a series of moves, where a move is conjugation by an
element of $S\cup S^{-1}$ or appending a relator. This construction can
clearly generate a list of reduced proof words – all 1-move proofs, all 2-move
proofs, etc. – but it is not obvious that it enumerates all trivial words.

######  Theorem 1.

Any reduced proof word $U$ can be generated from $\varepsilon$ by a series of
relator append and conjugation moves.

###### Proof.

Let $v_{1}$, $v_{i}$ ($2\leqslant i\leqslant N$) and $v_{N+1}$ be the freely
reduced forms of the words $u_{1}^{-1}$, $u_{i-1}u_{i}^{-1}$ and $u_{N}$ in
(1) respectively. Let $V$ be the product $v_{1}v_{2}\cdots v_{N+1}$. Note that
$V$ freely reduces to $\varepsilon$, but we do not freely reduce it at this
stage. Now consider the product $V^{-1}U$, which equals

(2) $v_{N+1}^{-1}\cdots v_{2}^{-1}v_{1}^{-1}\;v_{1}\,r_{1}^{\pm
1}\,v_{2}\,r_{2}^{\pm 1}\cdots v_{N}\,r_{N}^{\pm 1}\,v_{N+1}.$

If the subword $v_{1}^{-1}v_{1}$ is cancelled, then the remaining word can be
generated by a series of moves: start with $r_{1}^{\pm 1}$, conjugate by
$v_{2}$, append $r_{2}^{\pm 1}$, etc. Since $v_{1}^{-1}$ has been cancelled
from $V^{-1}$, the remainder $v_{N+1}^{-1}\cdots v_{2}^{-1}$ must freely
reduce to $v_{1}$. So we have generated the reduced proof word $U$, as
required. ∎

If a freely reduced word $T\in F$, called the _target_ word, is trivial in $G$
then there exist reduced proof words proving this. We can, in principle, find
these by enumerating the trivial words and checking them to see if they equal
$T$. (In fact, any conjugate of $T$ or $T^{-1}$ would suffice.) If we only
require one reduced proof word, say $U$, we can focus our search by attempting
to reduce $T$ to $\varepsilon$. Observe that the word $T^{-1}U$ must freely
reduce to $\varepsilon$, let $V$ be as in the proof of Theorem 1, and consider
the product $T^{-1}UV^{-1}$, which equals

(3) $T^{-1}\;v_{1}\,r_{1}^{\pm 1}\,v_{2}\,r_{2}^{\pm 1}\cdots
v_{N}\,r_{N}^{\pm 1}\,v_{N+1}\;v_{N+1}^{-1}v_{N}^{-1}\cdots v_{1}^{-1}.$

If we now cancel the subword $v_{N+1}v_{N+1}^{-1}$ and conjugate, we obtain
the expression

(4) $v_{N}^{-1}\cdots v_{1}^{-1}\;T^{-1}\;v_{1}\,r_{1}^{\pm
1}\,v_{2}\,r_{2}^{\pm 1}\cdots v_{N}\,r_{N}^{\pm 1}.$

This word can be generated by starting with $T^{-1}$ and then making a series
of conjugation and append moves. If the word (4) freely reduces to
$\varepsilon$ then we have generated a proof that $T$ is trivial in $G$. Since
any conjugate of $\varepsilon$ is trivial, the reduced proof word $U$ can be
recovered by conjugation, noting that $v_{N}^{-1}\cdots v_{1}^{-1}$ freely
reduces to $v_{N+1}$.

In the sequel we consider the case of proving that $E_{5}$ can be written as a
product of fourth powers in a 2-generator group. So we fix $S=\\{a,b\\}$, and
adopt the case inversion convention that $A=a^{-1}$ and $B=b^{-1}$. Our
relators are fourth powers of freely and cyclically reduced words (called
_base words_), and are finite in number. We assume that the set of relators
has been symmetrized; that is, it includes all inverses and rotations of the
words in $R$. If the 5th Engel word $[[[[[a,b],b],b],b],b]$ is expanded out
and freely reduced, then a word of length 72 is obtained. This word is of the
form $BBBB\cdots bbbb$, so we use as our target word the cyclically reduced
version, of length 64, and conjugate the resultant proof words by $bbbb$
(which is not treated as a relator).

Our base words are drawn from lists of bracelets on the generators $S$.
Bracelets are equivalence classes of strings under rotations and reversals. As
we are working within a free group (our alphabet is $\\{a,A,b,B\\}$), we use
the inverse of a word instead of its reversal and require our bracelets to be
freely and cyclically reduced. We call the resulting bracelets _reduced
bracelets_ , and modifying an extant bracelet enumeration algorithm to
generate these is described in [4]. The maximum length of the base words in
the proof in [3] is ten, and in Table 1 we list the counts of reduced
bracelets on two generators up to length ten. We also include the counts of
the Lyndon words, which are the reduced bracelets which are not proper powers.

Table 1. Base word counts (2 generators) word length | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---
reduced bracelets | 2 | 4 | 6 | 13 | 26 | 66 | 158 | 418 | 1098 | 2968
Lyndon words | 2 | 2 | 4 | 9 | 24 | 58 | 156 | 405 | 1092 | 2940

##  3\. Results

Havas’ proof in [2] was derived from the workings of coset enumerations using
a presentation for $B(2,4)$ with ten relators, using base words with lengths
from one to four. The presentation was a nine relator presentation for
$B(2,4)$ plus an extra relator to improve the performance of the coset
enumerations. A range of proofs was obtained by varying the settings of the
enumerations, with the shortest proof word found having 250 fourth powers.
Note that coset enumeration only uses the relators given in the presentation,
so powers not in the presentation do not appear in the proofs produced.

Korlyukov’s proof in [3] was obtained using a combination of computer and hand
calculations. He made use of the recursive nature of the definition of $E_{n}$
to rewrite $E_{5}$ in a shortened form using the subword
$a^{-1}b^{-1}aba^{-1}ba$. It is copies of this word or its inverse, combined
with $b^{n}$ for some $n\in\\{\pm 1,\pm 2,\pm 3\\}$, which yield the base
words of lengths eight, nine and ten in his proof word. The proof proceeds by
the judicious introduction of freely trivial words which yield fourth powers
in the rewritten word. This was continued until $E_{5}$ had been rewritten as
an element of the normal closure, yielding a proof word with 28 fourth powers.
Observe that there is no initial presentation. However an ex post presentation
is implicit in the proof word, and is simply the set of distinct relators
therein.

To facilitate proof verification and the comparison of proofs, we put all
proof words into a standardised reduced form. The relators are written out in
full and delimited by parentheses, with the strings of conjugating elements
outside the relators freely reduced. The parentheses are not part of the
proof, but allow us to distinguish between the conjugation and the relators.
If possible, matching generator plus inverse pairs which border a relator are
folded into the relator, reducing the length by two symbols and rotating the
relator (for example, $a(babababa)A$ can be replaced by $(abababab)$).

Table 2. Reduced proof word statistics | GH | AVK | CR |  xxxx1to4 | inf
---|---|---|---|---|---
overall length | 3180 | 616 | 444 | 716 | 842
count of relators | 250 | 28 | 26 | 48 | 60
sum of relator lengths | 1912 | 408 | 272 | 440 | 552
mean base word length | 1.91 | 3.64 | 2.62 | 2.29 | 2.30
conjugating pairs | 384 | 76 | 60 | 90 | 85
pairs per relator | 1.54 | 2.71 | 2.31 | 1.88 | 1.42
distinct relators | 10 | 12 | 13 | 11 | 12
group order | $2^{12}$ | $2^{24}$ | $2^{13}$ | $2^{13}$ | $\infty$

The statistics of all the proof words we discuss are given in Table 2, with
the proofs from [2] and [3] given in the “GH” and “AVK” columns respectively.
The overall length includes the parentheses. Korlyukov’s proof is much shorter
than Havas’ but has longer base words and more conjugation per relator. All
ten of the relators in Havas’ presentation appear in the proof word (this need
not be true in general), while Korlyukov’s proof has twelve distinct relators
and these present a finite group of order $2^{24}$ (that is, $|B(2,4)|^{2}$).

Using our reduction technique with random selections of base words from the 17
Lyndon words up to length four, we were able to generate a proof word with 48
powers. We do not record this proof word here, but give its statistics in the
“1to4” column of Table 2. It is significantly shorter than Havas’ original,
and its eleven relators present a finite group of order $2|B(2,4)|$. Repeating
this procedure using the 41 Lyndon words up to length five, we were able to
generate the proof word given below.

BBBBAbabABa(bAbAbAbA)(aBaBaaBaBaaBaBaaBaBa)Ab(AbAAbAAbAAbA)aBaaBa
(bAAbAAbAAbAA)(aaBaaaBaaaBaaaBa)AbAAAb(AAAA)(abABBabABBabABBabABB)
(bbbb)BBaBAb(baBAbaBAbaBAbaBA)abABabA(BBBB)b(bbabAbbabAbbabAbbabA)
aBAB(BaBABaBABaBABaBA)abAbabA(bbbb)B(BBabABBabABBabABBabA)aBAbbaBA
(bbbb)(BBaBBaBBaBBa)A(bbbb)B(AAAA)b(AAAA)aa(aaBaaBaaBaaB)bA
(AbAbAbAb)BaB(aBAbaaBAbaaBAbaaBAba)ABabAAB(aaaa)A(AAbAAbAAbAAb)
B(aaaa)(ABABABAB)babaaBAbabABabAbaBABaBAbabABaBAbaBABabbbbb

This proof word has thirteen distinct relators, and these present a finite
group of order $2|B(2,4)|$. Its statistics are given in the “CR” column of
Table 2. It contains 26 fourth powers and uses base words of all lengths from
one to five. Although 26 is only a slight improvement on Korlyukov’s 28, the
proof word is substantially shorter, due mainly to the shorter base words and
the reduction in the number of conjugating pairs. Attempts to generate a
shorter prof word by using longer Lyndon words (or reduced bracelets) did not
succeed.

The large group in Korlyukov’s proof begs the question: Can we find a set of
base words whose fourth powers present an infinite group in which $E_{5}$ is
trivial, and generate a proof therefrom? Augmenting the procedure used to
generate the “CR” proof with a check to accept only infinite presentations, we
were able to generate the required base words. The statistics for an example
proof word are given in the “inf” column of Table 2.

Our results demonstrate that, to generate short proofs, it is not necessary to
use base words up to length ten or to include base words that contain ‘long’
subwords of $E_{5}$. Nor is it necessary for the group presented to be
$B(2,4)$, or even finite. The most important factors seem to be the selection
of the base words and the inclusion of redundant relators (that is, relators
which are derivable from the other relators). For example, in the
presentations implicit in the “GH”, “AVK” and “CR” proof words, the 10, 12 and
13 distinct relators (resp.) can be reduced to 9, 8 and 8 (resp.) without
altering the group presented.

## References

  * [1] Burnside, W. (1902). On an unsettled question in the theory of discontinuous groups. _Quart. J. Pure Appl. Math._ , 33:230–238.
  * [2] Havas, G. (1981). Commutators in groups expressed as products of powers. _Comm. Algebra_ , 9(2):115–129.
  * [3] Korlyukov, A. V. (1993). Representation of a 5-fold commutator as a product of 4-th powers (in Russian). _Vopr. Algebry_ , 6:89–92. Zbl: 0925.20045.
  * [4] Ramsay, C. (2017). Listing words in free groups. _arXiv:1706.08188v1 [math.CO]_ , https://arxiv.org/abs/1706.08188.