PROCEEDINGS OF THE BURNSIDE WORKSHOP
BIELEFELD,
June-July
20-04,
20F50
1977, 2ri-230.
APPLICATION OF COMPUTERS
TO QUESTIONS LIKE THOSE OF BURNSIDE
GEORGEHAVAS AND M,F. ~JEWMAN
Computers have been used in seeking answers to questions related to those
about periodic groups asked by Burnside
in his influential paper of 1902.
A survey is given of results obtained with the aid of computers
program which manipulates
presentations
and a key
for groups of prime-power
order is
described.
I.
Computers
are becoming
Introduction
increasingly
important in studying questions
like those about periodic groups raised by Burnside
The actual questions
proceedings.
asked by Burnside are discussed
individual relations
in the paper Problems
in these
We consider
about a wider class of groups whose description may include
and residual finiteness
simplicity we refer to all these questions
are described
in his influential paper of 1902.
He was concerned with what are now called Burnside groups.
here similar questions
about groups
in addition to an exponent law.
as Burnside questions.
For
The main results
in section three below.
This report has its origins in a lecture given by one of us at the workshop.
Since then both of us have lectured a number of times on this theme.
report on this subject in the lecture "Computers
There is a brief
in Group Theory" given by Marshall
212
George Havas and M.F. Newman
Hall to the Galway Summer School in 1973 (see M. Hall 1977, pp. 34-35).
The growing impact of computers derives from their capacity for fast accurate
symbol manipulation on a grand scale.
The symbol manipulation procedure relating to
groups for which computers have been most extensively used is coset enumeration.
A
detailed account can be found in a paper by Cannon, Dimino, Havas and Watson (1973).
It has been widely used in the context of Burnside questions.
finite set
X : {a, b}
For instance, given a
and the finite set
R = {a 4, b 4, (o/p) 4, (a-lb) 4, [a2b] 4, [ab2) 4, r[a2"2'4D
J , [],a-lb-lab, 4,
of group words on
and having
R
as a defining set of relators has order
permutations on
1100/42
X , coset enumeration y i e l d s that the group
4096
G generated by
4096 = 212
X
and a pair of
letters which generate a group isomorphic to
G .
On a Univac
using the coset enumeration program, Todd-Coxeter V2.2, this result can be
obtained in about 30 seconds.
4 .
[a-lbab)4}
Hence
B(2, 4)
From the output it can be deduced that
has order
212
This example shows, in a simplified
form, the role of eoset enumeration in the study of groups.
G\S
representation of
G
of cosets of a subgroup
by permutations of
a finite generating set for
S .
has exponent
and has a presentation on two generators which
has a defining set of only nine fourth powers.
possible, the set
G
G\S
S
This is to find, where
in a group
G
and a
given a finite presentation for
G
and
Further examples of results obtained using coset
enumeration are given in section three.
In the context of the Burnside questions good progress has been made using
computers to perform manipulations with descriptions of groups of prime-power order.
For example,
1974),
B(2, 4)
cannot be defined hy fewer than nine fourth powers (Macdonald
B(2, 5) , the largest residually finite quotient of
(Havas, Wall and Wamsley
1974),
B(4, 4)
has order
2422
B(2, 5) , has order
and
5 $4
A , the largest
residually finite quotient of
A :: <x, y
has order dividing
2205 .
results were obtained.
result;
I m2 : y4 : i, exponent 8) ,
In section three we will give an outline of how the latter
Recently the same method has been used to obtain a related
see the note by Alford and Pietsch in these proceedings.
The programs used
are based on algorithms for calculating information about nilpotent quotients of
groups given by finite presentations and finite sets of identical relations or laws.
There have been a few rather brief accounts of such nilpotent quotient programs
published (Macdonald 1973, 1974, Wamsley 1974, Newman 1976).
While these describe
applications to Burnside questions, none of them goes into sufficient detail to
discuss a key point in such applications which is how to make law enforcement
practical.
This point is taken up in section two of this account.
213
Application of computers
Computers have also been programmed to perform other symbol manipulation
procedures related to groups.
Two such procedures which are playing an important role
in some current investigations are the Reidemeister-Schreier algorithm for calculating
presentations for subgroups of finite index in finitely presented groups (Havas 1974)
and manipulations
such as ~ielsen and Tietze transformations with presentations.
The programs we use are written in a readily portable superset of 1966 Standard
FORTRAN.
Further information on both programs and results can be obtained from the
authors.
We are indebted to many colleagues,
discussions and access to programs.
2.
too many to name individually,
for helpful
We thank them all collectively.
A nilpotent quotient program
In this section we describe a computer program, the Canberra nilpotent quotient
program, which, with its forerunners, played an important role in many of the
applications to be discussed in the next section.
The program provides the capacity
to manipulate presentations for groups of prime-power order efficiently.
it possible to study many groups of order
p < i000
,
n < 20 , and selected groups with
occasionally even higher.
kind;
pn
for moderate values of
n
This makes
p
and
n , say
well into the hundreds and
There have been essentially two previous programs of this
one produced by Macdonald (1974) and one by Bayes, Kautsky and Wamsley (1974).
The Canberra program has its origins in the latter and incorporates some important
features from it.
The core of the program is a set of routines for manipulating what we call powercommutator presentations.
prime-power order.
These are finite presentations which present groups of
Moreover every group of prime-power order has such presentations.
A power-commutator presentation
consists of a finite set
generators and a finite set of
n(n+l)/2
n
~i =
II
A : {a I, ...,
an}
of
relations
l)
a~ (i"
•
/:i+i
(2.l)
n
~(i ,j,1)
aja i = aia j ~
a
, i < j ,
l=j+l
where
p
is a prime and
a(i, 1), a(i, j, l)
be convenient to denote this presentation
< n; p, a) .
belong to
(n; p, a)
{0, i, ..., p-l}
.
It will
and the group it presents
The way in which such presentations may be thought of as occurring will
be explained later.
The word "commutator" is used because in most contexts the second
type of relation is written with a commutator on the left;
it is convenient here to
defer this change a little.
It is easy to see that every element of
< n; p, a)
can be represented by a word
214
George Havas and M.F. Newman
n
~ - [ a ~~ (1)
of the form
with
~(1)
in
{0, i . . . . .
p-l}
we call such a word normal
;
/=i
and abbreviate it to
~ , and use
~
for the identity element.
At the heart of this
whole approach is the simple fact that the given set of relations makes it straightforward to calculate a normal word equivalent to the product of two normal words.
It
will be convenient to discuss this (collection) process in the wider context of all
semigroup words on
A ;
inverses play little role in the present discussion.
that the set of left sides of the relations in
minimal non-normal semigroup words on
(n; p, ~)
(2.2)
A .
(n; p, a)
Notice
is precisely the set of
A collection process relative to
is a procedure of the following kind:
Given a semigrou p word
W
w
on
A ,
1
if
2
replace a minimal non-normal subword
is normal, stop~
the relation whose left side is
u
of
w
by the right side of
u , and return to
It is routine to prove that collection processes terminate,
1 .
in a normal word.
are many collection processes because of the choice in step 2.
There
A detailed discussion
of collection in the Canberra program is given in a paper by Havas and Nieholson
(1976).
The rest of this account is based on the collection process in the Canberra
program;
it would apply equally well to any other (fixed) collection process.
normal word resulting from applying this collection to a word
(w)
w
The
will be denoted
.
The first question which arises is:
number of normal words is
p
n
what is the order of
, the group
Unfortunately it often happens that
( n; p, a)
(n; p, a)
[~(~)]
.
whether the order of
has order at most
has order less than
case it is easy to see that there are normal words
((~)~] #
(n~ p, a) ?
6, ~, ~
p
pn
Since the
n
In this
such that
This leads to the following straight-forward test for determining
( n; p, ~}
pn
is
Let
T n be the set of words consisting of
all
akaja i ,
i e j e k ,
~ja i , a ja~ , i e j ,
a~.+I .
Observe that each word in
these overlap.
pn
has exactly two minimal non-normal subwords and that
Hence each word in
non-empty normal words and
order
L
~n, n~
T
n
can be uniquely written
minimal non-normal.
if and only if for each word
~
in
L
'
~
The group
I(~)~!
with
~, n,
< n; p, ~)
= [~(D~)I
has
If this
Application
is satisfied
is said to be consistent.
(n, p; a)
215
of computers
If
(n; p, a)
is not consistent,
then some test word will collect to two different normal words.
This leads to another basic procedure,
presentation
(n; p, a)
subset of (n-i
by
(n; p, a)
however,
and a pair
elements of)
and
A
p = v .
which, given a power-commutator
of distinct normal words, produces on a
a power-commutator
It is not difficult
presentation
discussion.
Before doing so, we complete our consideration
resulting
B, v
come from collecting a word in
power-commutator
presentation
presentation
for
<n; p, a) ;
of order
p
form.
aI
all the relations with
G
aI
there is a greatest positive
~, ~
.
presentation will be taken in
@
~
is obviously central and
$
(that is,
on the right.
~(1) # 0 ),
We only need a
k
and
~ = 9 .
such that
Since
z(k) # v(k)
~, w
.
p .
are distinct
Let
q(1)
be the
~'
be the
integer such that
q(1)(~(k)-v(k)]
The relation
<n; p, ~>
are obviously central and of order
(n; p, ~)
integer
to
can be read off from this.
which occurs in
on the left have
be the group defined by
leastnon-negative
(n; p, a>
in a power-commutator
reduction procedure when the words
of the order question.
the group defined by the
Let us say that a normal word
if, for every generator
in general;
for the present
to a consistent power-commutator
the order of
From now on the relations
power and commutator
L
is clearly isomorphic
Repeating this process leads, eventually,
for the group defined
to describe reduction
it is easier to describe a special case which suffices
When the words
Let
reduction,
p, ~
~ = ~
~ w(1) - p(1) modulo p .
is equivalent to
k-i a~(1 )
(2.3)
[with
ak
aI
obviously
:
central and of order
p
whenever
Q(1) # 0 ].
Let
function defined by
~'(i, l) E ~(i, l) + q(l)~(i, k) modulo p ,
a'(i, j, l) ~ a(i, j, l) + q(1)a(i, j, k) modulo p ,
and
0 ~ a'(i, l) ,
Then
all
G
is defined by
(n; p, ~')
i, j , the generator
ak
a'(i, j, l) < p .
and 2.3.
Since
and when it does the relation is a consequence
ak .
Hence the presentation
involving
G
on
ak
n - i
and
ak
obtained from
from the generating
generators;
a'(i, k) = a'(i, j, k) = 0
only occurs on the left of relations
in
for
(n; p, ~')
;
of 2.3 and relations not involving
(n; p, a')
by deleting all relations
set is a power-commutator
this is what was required.
presentation
for
216
George Havas and M.F. Newman
Clearly it would be desirable
from the point of view of computational
to make the set of test words for consistency
redundancies;
for example,
further structure
reduction
one never needs to test
into power-commutator
presentation
(2.4)
it practical
has exponent
pf
(ii)
presentation
some of its relations
(a)
(b)
will be called w e i g h t e d
~
from
~(i) = I
and
for each
k
~(i) ~ ~(i+i)
with
~(k)
> i
there is exactly one definition
ak
is
aPi = ak
(d)
if the definition
of
ak
is
~aj, a t
: w(i) + ~(j)
if
~(i, l) ~ 0 , then
if
a(i, j, l) # 0 , then
these is the definition
of
ak .
~(1)
weight
(n; p, a)
~(n)
= ~(i) + i ;
= a k , then
~ ~(i) + i ,
~(1) e w(i) + ~(j)
ak
function;
kind occu~ in the course of the calculations
integer
w(k)
.
as right side but only one of
It is easy to see that there are power-commutator
which admit no such
presentation
, then
;
Note there may be more than one relation which has
The positive
to the positive
ak ;
of
group it defines.
n}
;
if the definition
~(k)
commutator
and
{i . . . . .
(c)
presentations
if
such that
whose right side is
(e)
structure also leads to a
a group given by a power-commutator
are called d e f i n i t i o n s ;
it admits a weight function
integers
some
.
A power-commutator
(i)
We now introduce
which leads to a useful
This additional
to test w h e t h e r
efficiency
There are some obvious
anaja i .
presentations
in the test set for consistency.
criterion which makes
smaller.
to be described.
with weight
is the class
of
indeed presentations
w
A weighted power-
will be denoted
(n; p, a; ~)
and of
This class is, see 2.6, the exponent-p-central
u s u a l l y d e f i n e d in terms of the lower exponent-p-central
of this
(n; p, a; ~)
{ n; p, ~; ~
class which
series of a group
.
, the
is
G ~
that
is, the chain
G = G O > ... > G h > G h
-
of subgroups
central
class
where
c .
-
Gh+ I = EGh, G]G~
.
-
If
+ l
> • ""
-
ac_ I > G c = E
, then
Note that a group with exponent-p-central
and has nilpotency
class at most
exponent-p-central
class often seems the more significant.
about the lower exponent-p-central
c .
O
class
has e x p o n e n t - p c
is nilpotent
In the study of groups of prime-power
series
is
An important
order the
observation
Application of computers
(2.s)
217
Ess' ~h] s ~+h+l "
This is proved in essentially the same way as the corresponding result for the lower
central series.
(2.6)
The exponent-p-central class of
Let
G = (n; p, ~; w) .
~(k) { h + 1 .
by all the
ah
This is clear for
with
[al, ai]
and all the
We prove that
w(1) ~ h .
with
h = 0 .
(n; p, a; ~>
Gh
For
ai
and
ak
a k = a~t with
then either
~(k) : w(i) + ~(j)
.
with
~(k)=
Gh
i
or
ak
with
is generated
is generated by all the
aT
arbitrary, and so, by clause (a) of
~(k) ~ h + i .
~(i)+
.
Gh_ 1
h { i , suppose
Gh
the definition of a weighted power-commutator presentation,
subgroup generated by all
~(n)
is generated by all the
It follows that
~(1) ~ h
is
is contained in the
Conversely, if
a k = ~j, ai]
~(k) { h + i > i ,
with
Hence, by the inductive assumption and 2.5,
ak
belongs to
Gh •
A first gain which comes from using weighted power-commutator presentations is
that the set
(2.7)
L
T*
of test words for consistency can be reduced to
Y*n :
is the set of all words
n
akaja i ,
with
~(i) + ~(j) + ~(k) ~ ~(n)
i < j < k ,
,
a ai, S ,
with
i + ~(i) + m(j) 5 ~(n)
i<5,
, and
a~.+I
with
i + 2~(i) < ~(n)
This is so because it is straight-forward to check that for every other test word
collection yields the same result;
[a[]ai = ai[a[]
since
is a useful reduction,
exceeds
3 ;
[a[]
for example, if
only involves
a I with
i + 2~(i) > ~(n)
~(/){
~(i)+
, then
i.
While this
in practice redundancy is usually observed once the class
moreover this redundancy increases with clas~.
It would be desirable to
have this observation reflected in still smaller test sets.
A second gain is a criterion which makes testing whether an exponent law holds
reasonably practical.
For a normal word
~ , let us call
~
/:i
6.
~(1)~(1)
the weight of
218
George Havas and M.F.
(2.8)
The group
<n; p, a; w)
normal words of weight at most
The criterion
by an argument
[Xl..• ~)/
8
PJr
has exponent
~(n)
(1959, p. 169;
it is not as conveniently
in the free group
xj
is the product
of commutators
criterion
F
put).
in
X
each of which
(ordered)
~
~(n)
weight of
N
homomorphism
then
, then clearly
~P
(X 1 . . .
F
to
XS) 6 : [
.
~ .
(n; p, e; ~}
subset of
X
and
X .
be a normal word.
Otherwise
assume
n
s = ~
$(1)
/:i
Let
which maps
i-1
i
/:i
/:i
Hence
..., Xs}
to give
u
is a product
From this the
If the weight of
is
/
= ~ .
is less than that of
from
(1967)
u
involves each element of
Let
The crux is covered
X : {Xl,
Y
at most
for all
It is applied to the word
freely generated by
of some proper
is proved by induction•
: ~
or see §3.3 of Hanna Neumann's book
•
where
~Pf
provided
.
can be proved by a simple varietal argument.
of Higman
where, however,
Newman
xk
D
= ~
•
Let
ai
to
whenever
8
the
be the
where
the above
by
= I I
(x~e)
.~o.
J
Clearly
xjO
(xj81pf
= ~ .
commutators
weight of
that
is a normal word and its weight
From the structure of
in
~
A
exceeds
u@ = ~ .
u
each with at least
Hence
This criterion
~(n)
$Pf
and
= ~
it follows
~(1)
quite practical,
has class
u8
is a product of
for each
~(n)
1 .
Since the
, it follows readily
into the program of Bayes, Kautsky and Wamsley
it).
Macdonald's
(1974) criterion seems
While the criterion makes testing w h e t h e r
the calculation
limitation to these explorations
restrictions
a1
~ , so
as required.
is incorporated
testing much redundancy
that
entries
<n; p, e; ~)
(though their paper (1974) does not m e n t i o n
to be of a similar kind.
is less than that of
exponent
laws hold
of even the powers specified by it is the most severe
of groups of Burnside type.
is observed,
so it would be desirable
on the powers that need to be calculated.
As with consistency
to find further
Applicaffon of compufems
In practice we have used other considerations
done.
In our initial work on
congruences.
B(4, 4)
219
to extend the scope of what can be
we used related commutator laws and
The relevant calculations are more complicated to program, but are then
quicker to execute.
The simplest example of this comes from the law
[y, x]2[y,
which holds in
B(2, 4)
criterion that
< n; 2, a; ~)
x, x, x][y, x, y, x]Ey, x, y, y] =
and therefore in all groups of exponent
has
exponent
4
provided
4 .
a~ = ~
This yields the
for all
k
and
[q, ~]2[n, ~, ~, ~][q, ~, q, ~][q, ~, q, n] =
for all normal words
~, q
such that
~q
is a normal word of weight at most
More complicated criteria come from allowing more variables.
quotient program can handle
B(4, 4)
~(n)
.
The present nilpotent
without such special considerations.
We hope,
if resources permit, to implement these more refined methods in such a way that they
will allow the routine testing of other laws such as commutator laws.
our main goal was to prove finiteness.
preimage of
~
is finite;
or, equivalently,
to find a set
that the residually finite group generated by
defining set of relators is finite.
In studying
For this it suffices to show that some
{x, y}
with
E
of eighth powers such
{x 2, y4} u [
as a
Therefore we used heuristic considerations to
limit the eighth power calculations done;
more details are given in the next section.
Just as reduction is used to make a power-commutator presentation consistent, so
it can be used to enforce exponent laws; that is, given
pf ,
a power commutator presentation for the largest exponent
pf
~Pf # ~
If testing shows that
of
(n; p, a)
~Pf
= ~ .
for some
to yield from
quotient of
(n; p, a)
< n; p, a)
pf
< , then the largest exponent
is the same as that of the group defined by
(n; v, a)
quotient
and
Applying reduction yields a power-commutator presentation for this group
which has fewer generators.
Repeating this process of testing and reduction clearly
leads to the required power-commutator presentation.
We introduce next the
commutator presentation
~(d) = i < ~(d+l)
The
p-covering presentation for a consistent weighted power-
(n; p, ~; ~)
, then
P
p-covering presentation
al, ..., ad+n(n+l)/2
(i)
the
(2)
d + n(n-l)/2
where
n - d
.
Let
P = < n; p, ~; ~)
can be gcnerated by
for
(n; p, a;
.
Suppose
d , and no fewer, elements.
~o) has a generating set
and relations of three kinds:
definitions
in
of the form
uk = V k
(n; p, ~; ~)
uk =
is a relation in
vkak
for
,
k E {n+l . . . . .
(n; p, ~; ~)
d+n(n+l)/2}
which is not a
.
220
George Havas and M.F. Newman
definition,
(3)
an+ I ..... ad+n(n+l)/2
those w h i c h specify that
order
are central and of
p .
Relations of the first two kinds are definitions.
Because
(n; p, a; ~)
is
consistent, if the c o n s i s t e n c y test fails for a test w o r d r e l a t i v e to the
p-covering
presentation, the r e s u l t i n g n o r m a l words are o b v i o u s l y central and of order
p
, and
so r e d u c t i o n can be used to y i e l d a c o n s i s t e n t p o w e r - c o m m u t a t o r p r e s e n t a t i o n
(n*; p, ~*)
n* - d
with
definitions.
into a w e i g h t e d presentation.
of
P .
P*
Clearly
~(n) + i
w h o s e largest class ~(n)
capable.
(n*; p, ~*) .
quotient is
P*
uk =
vk
otherwise the class is
p
P
P , it can be shown that
P*
is
a~ :
an+ 1
with
~(n)
and
P
~(i)
P*
as a group of
and
d
generators~
or
Q
and
is said to be
an+ I
is terminal.
and
in
P
with
is capable;
The kernel of the m a p p i n g from
H2IP, ~p]
P
.
of
P
p - p o w e r order, we m e a n that the g r o u p
<XIR)
is isomorphic to
P
{XI~}
G .
P*
over the field of
n* - n
The rank
of the
as a group of
defines a group .G
has a largest q u o t i e n t of
n* - n
It follows that
is a lower b o u n d on the number of r e l a t i o n s needed to define
P
as a
d generator
It r e m a i n s a n o p e n q u e s t i o n w h e t h e r this lower b o u n d is always attained.
P*
Since the class of
is at most
w(n) t i , it is p o s s i b l e to introduce fewer
P*
generators in w r i t i n g down a p o w e r - c o m m u t a t o r p r e s e n t a t i o n for
~(n) : i
It suffices to add a new generator for each r e l a t i o n in
w h i c h n e i t h e r is a d e f i n i t i o n nor has its left side of the form
~(i) + ~(j) ~ ~(n) + 2 .
possible.
For example,
if
unless
(n; p, ~
[aj, ai]
w)
with
This is of value b e c a u s e it means fewer r e d u c t i o n steps are
needed to reach a c o n s i s t e n t p o w e r - c o m m u t a t o r presantation.
and if
is a
~j, ai] : an+ I
re(n) + i
in saying a p r e s e n t a t i o n
p - p o w e r o r d e r and that this quotient
group.
P
is the least number of pelations needed to define
p - p o w e r order on
Q
, there is no such
Q
: ~(n)
p-multiplicator of
we call it the
p-multiplicator
~(n)
is
is isomorphic to the second h o m o l o g y group
elements~
is a group of class
can be d e t e r m i n e d from the d e f i n i t i o n of
If this is
p-covering group
is the
Q
which are not d e f i n i t i o n s are suitably ordered
~(i) + ~(j) = ~(n) + i , then the class o f
to
If
otherwise, there is such a
If the relations
then the class of
~(n) + i .
P* . If the class of
terminal;
is said to be
P* = < n*; p, a*)
The group
has class at m o s t
h o m o m o r p h i c image of
P
This p r e s e n t a t i o n can not in g e n e r a l be made
~(k) : 2
and
[aj, ai]
ak
Further cuts are
is the d e f i n i t i o n of
ak
~(1) = ~(n) - i , then
Eal,
ak] =
Eal, aj, ai] [al, ai, aj] p-I
To tell m u c h m o r e of this story requires technical details that it is not a p p r o p r i a t e
to go into here.
Note though that it implies that
~(i) = i
in all definitions of
Application
[aj, ai] = a k .
the form
We have now developed enough machinery
power-commutator
Bid, pf)
presentation
B(d, pf; c) .
which we denote
Bid , pf;
iI :
with relations
and
~j, ai]
= ¢
constant function with value
weighted power-commutator
down a power-commutator
p, ~*)
If
;
B(d, pf],
follow.
For example,
generators,
shows
to the presentation
needs
9
order;
well.
B[d, pf)
, of
a power-commutator
for
2-covering
on
for
be
hence
4th
is well-known
for
21
B(d, pf]
and
w'(i)
If
power
n' = n ,
has
B(2, 4)
has order
12
this latter
4
reduces
212 , class
powers, to define it as a group of
5
= c
for
i > n .
Kostrikin,
the procedure will eventually stop.
and
2-power
~'
on
(n'; p, a')
Thus this procedure
In practice
f = i
(for
In the study of groups, such as
we work with presentations
A , whose description
by a result of
p ~ 5 ) this has
which,
includes
presentation.
group given by the finite presentation
IbI, ..., bm; u I = v I ..... u r = Vr} ,
we work with augmented power-commutator
individual
in effect, combine the given finite
power-commutator
presentations
which have
by:
can be
B(2, 5) .
and an appropriate
it
to be finite, these conclusion, s apply to it as
is finite, as it is known to be when
happened so far only for
.
B(2, 4; 5)
generators;
can be extended to a weight function
i S n
pfth
and the other consequences
presentation
group has
(n'; p, ~')
is capable, however enforcing exponent
B(2, 4; 5) ;
~
- i
C
p , so the condition needed for reduction
(n; p, a; ~)
B(2, 4; 5)
is
is
pn , class
n* - n
has order
pf
If
presentation
d
and
(n; p, a; ~)
is capable, enforce the exponent
iterated.
relations,
If this shows
B[d, Pfl
Write
<n; p, ~; w>
and
is presented by
B(2, 4)
The function
~'(i) : ~(i)
~(n) : c - i .
generators
presentation
relations, which can be
since
with
be a consistent
group of
B(d, pf]
(consistent)
and that of its
presentation
and weight function the
c-l)
finite quotient,
adl
{al . . . . .
note that in this context the normal word collected from a
Let the resulting
then again
set
(n; p, a; ~)
(n*; p, ~*) .
will be obviously central and of order
holds.
i, j
p-covering
p-power order by
(n; p, a; ~)
generating
B[d, pf;
for
It follows that
and can be defined as a group of
powers.
for all
for the
presentation
(n; p, a; ~) .
it has a
c > i , let
presentation
then the largest residually
presented by
(n*;
For
presentation
reduce it to a consistent
terminal,
i .
quotient of the Burnside group
Begin with a consistent weighted power-
for
~/ = ¢
to outline how to determine a consistent
for the largest class c
commutator presentation
pfth
221
of computers
Thus in studying a
222
George Havas and M.F. Newman
{bl, ..., bm, al,
.... an}
as set of generators, and relations
n
Ul
with
p(h, l)
= Vl"
in
"''"
Ur
= Vr
"
{0, i, ..., p-l}
= ~ a
bh
1
l < h < m
'
-
-
, and all the relations 2.l.
Where relevanT, there is a weight function
however, now there is a definition for each
~o(d) = I < ~(d+l)
p(h'l)
bh
, have left side
~
as before on
a k , the first
for some
d
{i,
..., n} ;
definitions, where
h .
In this context every semigroup word on the generators can be collected to a
normal word on
{a I . . . . .
an}
by adding the steps:
Where the inverse of a generator
bm+ i
generator
{i,
= (Vk)
, where
..., r}
.
( )
bh
I I a~ (h'l)
by
is involved we deal with it by introducing a new
%
and including among the defining relations
presentation is consistent
(uk]
b.
replace
b .bm+i
= ~
if, in addition to the previous conditions,
denotes the result of collection,
for all
We say the
it satisfies
k
in
Thus, if an augmented power-commutator presentation is consistent, the
n
p
and is a quotient of the group given by the input
group it defines has order
presentation.
presentations.
We next modify the procedure for writing down
definition (nor has large weight left side).
n : 0 ;
set
Note this includes the case when
then there are no definitions and the
{b I . . . . , bm, a I . . . . , am}
"''" Ur
bk : a k ,
a.
p-covering presentation has generating
and relations
Ul : VI"
and each
p-covering
A new generator is added to each relation which is neither given nor a
is central and has order
:
V
i <- k <- m
r
,
p .
We can finally outline how to determine a consistent augmented power-commutator
presentation for the largest class
<b I . . . . .
bm
Begin with the largest class
c
quotient of
l u I = v I .....
0
quotient:
u r = Vr, exponent pf>
it has generating set
.
{bl, ..., bm}
and
relations
U I = Vl,
For
P
...~ U r = V r ,
b I : ~ .....
bm : ~ .
c ~ i , let there be given a consistent augmented power-commutator presentation
for the largest class
c - i
quotient.
it and reduce to a consistent presentation.
largest quotient of
Write down the
If this is
p-power order which is presented by
p-covering presentation of
P , the given group has a
P .
Otherwise enforce the
A p p l i c a t i o n of computers
exponent
pJ .
The same conclusion
223
follows if this reduces the presentation
to
P .
If not, extend the weight function as before.
3.
3.1
Results
EXPONENT FOUR
Leech (1963) was the first to publish results on Burnside groups obtained with
the aid of a computer.
and four.
He produced presentations
for some groups of exponent
For example he found, using computer coset enumeration,
nine fourth powers which suffice as relators
for
B(2, 4) .
three
various sets of
He also reported on a
group of Burnside type suggested by Philip Hall, namely
I3 = <a, b, c I a2 = b2 = c2 = ~, exponent 4> .
Leech computed the order of
With a view towards
I3
B(3, 4)
and found ten relator presentations
he investigated
for this group.
the group
I2 = (a, b, c I a2 = b2 = ~, exponent 4) .
Macdonald
(1973, 1974) used a nilpotent
about Burnside groups.
relations
I2
and for
has order
I3
He showed that minimal presentations
have ten relations,
219 .
quotient program to prove various results
for
as achieved by Leech.
Bayes, Kautsky and Wamsley
B(2, 4)
(1974) computed the order of
B(3, 4), 269 , and Alford, Havas and Newman (1975) computed the order of
2422 , in both cases using a nilpotent
appear) has used coset enumerations
in
quotient program.
B(2, 4)
have nine
He also showed that
B(4, 4),
More recently Havas (to
to express the fifth Engel word as a
product of fourth powers.
We present here some new results about various groups of exponent four and in
particular about
computations
I2
and about
B(4, 4) .
is
I2
has order
219 .
quotient program we readily confirm Macdonald's
We determine
20 , which shows that at least
involutory relations
nilpotent
18
that the rank of its
fourth power relations
are required to provide a presentation
quotient program also produces a list of
with the two involutory relations)
order.
of the
involved.
Using the Canberra nilpotent
result that
We include a description
is sufficient
18
for
presentation
list is quite suitable for nilpotent
other purposes,
to define
I2
short.
In the case of
The Canberra
as a group of
2-power
shortest words in
generators which suffice.
quotient calculations
such as coset enumeration,
usually particularly
I2 .
2
fourth powers which (together
This list is routinely obtained as the lexicographically
terms of the power-commutator
2-multiplicator
in addition to the
This kind of
but less satisfactory
for
because the free group word length is not
I2
one of the longest such words is
224
George Havas and M.F. Newman
lacac-lacbc-lacabl 4
, a free group word of length
12 , raised to the fourth power.
However other sets of fourth powers can be found readily.
calculate the
2-covering group of
covering group.
f2
One approach is to
and then to collect fourth powers in that
Collection of short powers (short in the free group sense) reveals
that the following relations suffice to define
f2
as a group of
2-power order:
a 2 = b2 = c 4 = (ab) 4 = (ac) 4 = (bc) 4 = (abc) 4 = (acb) 4 = (ace) 4 = (bcc) 4
= (abac) 4 = (abcb) 4 = ( a b c c )
4 = (acbc) 4 = lacbc-ll 4 = (ababc) 4 = ( a b c b c )4
= (bacac) 4 =
Ibc-lacal 4 = (bcaca) 4 = ~ •
Such short relations are much more suitable for coset enumeration.
enumerate the
8192
cosets of
<a, c>
It is easy to
in the group presented this way.
(The
corresponding coset enumeration in the group presented with fourth powers routinely
given by the Canberra nilpotent quotient program has not been successfully
performed.)
64
Since
<a, c I a2 = c4 = (ac)4 = ( a c c ) 4 = ~>
(very easy by coset enumeration)
a minimal presentation for
it follows that the above
Note that some of the claims in the literature about
<ab,
c>
group of order
is isomorphic to
210
20
relations provide
I2 .
(1963) gives the incorrect order for the group.
that
presents a group of order
B(2, 4)
I2
are wrong.
Thus Leech
This arises from the misapprehension
of order
212 .
In fact
, which follows from the above calculations.
<a b, c>
is a
This may also be
shown by explicit calculation with the power-commutator presentation for
I2 .
E. Oppelt of RWTH Aachen has implemented routines which augment the Canberra nilpotent
quotient program and provide facilities for subgroup construction and investigation by
computer, and which can be used for this purpose.
Leech (1967) endeavours to improve on the presentation for
12
and also refers
to an attempt by Sinkov at obtaining a presentation for that group (which is the
presentation given by Cannon (1974)).
different groups of order
221
The presentations of Leech and Sinkov define
which have
I2
as a quotient.
Using the Canberra nilpotent quotient program we have computed the rank of the
2-multiplicator of
B(3, 4) .
This rank is
fourth powers which suffice to define
i08
B(3, 4)
and we have found a set of
as a group of
105
2-power order.
A very significant computational effort was put into determining a consistent
power-commutator presentation for
B(4, 4) .
The initial motivation was to gain
information on the solubility question for groups of exponent four.
At the time that
the computations were first performed it seemed possible that the determination of
B(4, 4)
could perhaps be combined with results of Gupta and Newman (1974) to prove
the solubility of all groups of exponent four, if that were in fact the case.
Net
surprisingly this did not succeed because insoluble groups of exponent four do exist,
Application
as was proved by Razmyslov
detailed information
presentation
B(d, 4)
available on
However Vaughan-Lee
(1979) has used the
B(4, 4) , via the consistent power-commutator
computed for it, as a basis for determining
explicitly
technical
(1978).
225
of computers
in terms of
d .
As mentioned
the solubility
innovations were made for the initial computation
significant
collection.
of
B(4, 4) .
saving in computer time was made by the use of formulas
Whereas the Canberra nilpotent
both to process the consistency
initial run for
B(4, 4)
length of
in section two, a number of
A
instead of direct
quotient program uses direct collection
equations and to perform exponent testing,
commutator
in the
formulas were calculated and used instead, where
which
correspond to consistency
possible.
Thus formulas for Jaeobi identities,
equations,
and for fourth power expansions were computed.
Substitution
in this type
of formula is superior to direct collection because the bulk of collection required
has been done once and for all in computation
computer technology
nilpotent
quotient program,
to compute
nilpotent
B(4, 4) .
.
B(4, 4)
of the formula.
With improvements
in
of the latest version of the Canberra
formula substitution methods
are no longer needed in order
Indeed we have used the current version of the Canberra
quotient program to recompute
2-multiplicator,
1100/42
and the development
namely
1055
.
Note that the rank of the
requires at least
1055
B(4, 4)
This calculation
and also to compute the rank of its
took about
2-multiplicator
relations.
power relations which suffice to define
i0
hours on a Univac
shows that a presentation
We have computed a set of
B(4, 4)
as a group of
for
1055 fourth
2-power order.
Other groups of exponent four for which we have compute& consistent powercommutator presentations
are:
(a, b, c I a2 = ~, exponent 4) , which has order
237
and class
7 ;
(a, b, c, d I a2 = b2 = c 2 = d 2 = ~, exponent 4) , which has order
class
and
5 ;
(a, b, c, d I a2 = b2 = c2 = ~, exponent 4> , which has order
class
238
266
and
6 ;
a, b, c, d I a2 = b2 = ~, exponent 4> , which has order
2120
and class
8 ;
a, b, c, d I a2 = ~, exponent 4> ~ which has order
2224
and class
i0 ;
(a, b, c, d, e I a2 = b2 = c2 = d2 = e2 = ~, exponent 4) , which has order
2138
and class
6 ;
a, b, c, d, e I a2 = b2 = c2 = d2 = ~, exponent 4) , which has order
and class
7 .
2228
226
3.2
George Havas and M.F. Newman
EXPONENT FIVE
The order of
B(2, 5)
was shown by Havas, Wall and Wamsley
using computer calculations
computations
in two different ways, one via nilpotent
and the other using Lie algebraic methods.
obtained a consistent power-commutator
Canberra nilpotent
quotient program.
the order of the free
namely
5926
S-generator
This provides
Calculations
being performed using nilpotent
and presentation manipulation
3.3
for
B(3, 5; 9)
has order
5916
the
confirmation
using the
which is less than
4th
Engel condition,
of a result of Wall (1974)
including M. Hall Jr., G. Havas, J.S. Richardson
the unrestricted
problem for exponent five with
in search of a finiteness
quotient,
proof for
coset enumeration,
B(2, 5)
are
subgroup presentation
programs.
a consistent power-commutator
This is a group of order
3.4
quotient
EXPONENT SEVEN
We have constructed
b,
B(3, 5; 9)
an independent
are investigating
computer assistance.
presentation
534
More recently we have
Lie algebra satisfying
Currently a number of people,
and J. Wilkinson,
(1974) to be
8x] # ~
, so the
7668 .
8th
Engel
If
x
and
y
presentation
are generators
for
B(2,
7; 13) .
of this group then
identity does not hold.
EXPONENT EIGHT
Detailed results about a number of groups of exponent eight are presented
other papers in these proceedings•
details about one particular
Here we present some results and computational
group of exponent
In the paper by Grunewald,
8 .
Havas, Mennicke and Newman in these proceedings
problem as to whether the subgroup
(a 4, b 2)
of
B(2, 8)
is finite is posed.
show that it has a largest finite quotient by showing that a residually
preimage
Let
in
the
We
finite
is finite.
A := (a, b
I a2 = b4 = ~, exponent 8) .
has a maximal nilpotent quotient of order
2205
Then
£
has a preimage
and class
26 .
D
which
This preimage was
found by using the exponent testing procedure
to partially enforce the exponent law.
The class
in the normal way.
i0
quotient of
A
was calculated
class the exponent tests were taking more and more time.
two, the exponent law test has a lot of redundancy
the class
i0
quotient of
powers of
o/~
and
~b
A
so that, for example,
to compute
only two eighth powers are actually required,
which are words of weight
may suffice to test only words of low weight.
2
and
From class
3 .
i0
testing was performed only for words with weight less than
defined by two generators,
With increasing
As pointed out in section
one of order
2
and one of order
eighth
This suggests
that it
onwards eighth power
ii .
D
is the group
4 , in which all normal
Application
words of weight less than
2-power order
weight
i0 .
consistent
69
ii
have order dividing
further exponent
power-commutator
presentation
8
strongly suggests
that the nilpotent
2-multiplicator
for the nilpotent
18
quotients of
quotient of
D .
have order dividing
A
and
D
of the maximal nilpotent
It was
8 .
This
are in fact the same.
quotient of
D
is
69 .
EXPONENT NINE
We have computed a consistent
which is a group of order
Macdonald
3
To define this as a group of
checking was applied to the
found that all normal words of weight less than
3.5
8 .
eighth powers are needed, with those of highest weight having
Subsequently
The rank of the
227
of computers
presentation
for
B(2, 9; 12)
3724
(1973) points out that "even groups generated by two elements of order
lead to depressingly
large results",
Macdonald showed that the group
I a3 = b3 = (ab)3 = (aab)9
and this is confirmed by Shield (1977).
a 3 = b 3 = (ab) 3 = ~, exponent 9>
<a, b
largest finite quotient of order
<a, b
power-commutator
243 .
= ~>
has a
It is easily shown by coset enumeration
zs a presentation
that
for this finite group, and in
fact a minimal presentation.
By applying the Canberra nilpotent
quotient program we have shown that the
restricted Burnside problem is answered affirmatively
9
generated by two elements of order
nilpotent
3
for the freest group of exponent
whose commutator has order
3 .
Thus the
quotient program shows that a preimage of
(a, b I a3 = b3 = [a, b] 3 = ~, exponent 9)
has a largest finite quotient of class
The unrestricted
class
371
(a, b I a3 : b3 : [a, b] 3 = (ababb) 3 : ¢, exponent 9) .
The
quotient program shows that this group has a largest finite quotient of
i0
powers
and order
problem has been solved for a quotient of the free group
considered above, namely
nilpotent
18
and order
312 .
The nilpotent
((06))9 , (az~b)9 , (aabd~b) 9)
quotient program provides
which suffice to define this group as a group of
3-power order when added to the four explicitly given third powers.
the cosets of
<a,
yields index
b
(d69)
Enumeration
of
in
I a3 = b3 = [ a ,
59049
three ninth
b ] 3 = (ababb) 3 = ( a b ) 9 = ( a a ~ ) 9 = ( a a b a b ) 9 = ¢ )
, completing the solution of the unrestricted
problem in this
case.
We have applied the nilpotent
generated by two elements of order
quotient program to the freest group of exponent
3
whose product has order
date reveal that it has a preimage whose class
18
3 .
Computations
quotient has order
3603
to
9
228
George Havas and M.F. Newman
References
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Notices Amer. Math. Soc. 22, A.301.
A.J. Bayes, J. Kautsky and J.W. Wamsley (1974), "Computation in nilpotent groups
(application)", Proc. Second Internat. Conf. Theory of Groups (Canberra,
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MR50#7299;
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Zbi.288.20032;
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John J. Cannon, Lucien A. Dimino, George Havas and Jane M. Watson (1973),
"Implementation and analysis of the Todd-Coxeter algorithm", Math. Comp. 27,
463-490.
MR49#390;
Zbi.314.20028;
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MR50#7300.
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Comm. Algebra.
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George Havas and Tim Nicholson (1976), "Collection",
Symbolic and Algebraic Computation,
229
SYMSAC '76 (Proc. ACM Sympos. on
Yorktown Heights, New York, 1976), pp. 9-14
(Association for Computing Machinery,
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MR51#3298;
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