Article Submitted to Journal of Symbolic Computation
Constructing faithful representations of
nitely-generated torsion-free nilpotent
groups
Willem A. de Graaf1 and Werner Nickel2
1
Mathematical Institute, University of Utrecht, PO BOX 80.010, 3508 TA
Utrecht, The Netherlands
2 Fachbereich Mathematik, TU Darmstadt, Schlogartenstrae 7, 64289
Darmstadt, Germany
Abstract
We formulate an algorithm for calculating a representation by unipotent
matrices over the integers of a nitely-generated torsion-free nilpotent
group given by a polycyclic presentation. The algorithm works along a
polycyclic series of the group, each step extending a representation of an
element of that series to the next element.
1. Introduction
A nitely-generated, nilpotent and torsion-free group is called a T -group. An example of a T -group is the subgroup Ur (Z) of the general linear group GL(r Z)
consisting of all upper-triangular matrices with all diagonal entries 1: By a theorem of S. A. Jennings (see 5]) every T -group can be embedded into Ur (Z): In this
paper we consider the algorithmic problem of constructing such an embedding
from a polycyclic presentation of a T -Group.
Lo and Ostheimer (8]) published a rst algorithm for this purpose. For a T group G they dene a right ideal I of the group ring ZG such that ZG=I is a
nite-dimensional free Z-module, and a faithful G-module. In order to construct
a basis of this quotient the algorithm relies on the calculation of a Grobner basis
for I:
The main idea of our algorithm is borrowed from a proof of Ado's theorem
for Lie algebras, as can for instance be found in 1], see also 4]. We work our
way up a polycyclic series of G. The basic step consists of an algorithm for
extending a unipotent matrix representation of a T -group N to a unipotent
matrix representation of a T -group H , where N is a normal subgroup of H ,
and H=N = Z. We let H act on the dual space (ZN ) and construct a nitedimensional faithful submodule.
1
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 2
As a byproduct this also leads to an essentially new proof of the theoretical
result that every T -group can be embedded into Ur (Z): Our algorithm uses only
simple concepts from linear algebra. As a consequence, its implementation is
less involved than the Lo-Ostheimer approach. Moreover, experiments suggests
that the algorithm presented here produces matrix representations of smaller
dimension.
This paper is organized as follows. Section 2 recalls properties of T -groups
needed later. In Section 3 we give the theoretical foundation of our algorithm. In
Section 4 we formulate the algorithm and illustrate it with an example. Finally,
in Section 5 we discuss the implementation of the algorithm in the computer
algebra system GAP 4 (3]), and the running times of the program on some
sample inputs are given.
2. Preliminaries
A T -group is polycyclic. In the following we will recall basic facts about polycyclic groups, specialised for the context of T -groups. Sims 10, Chapter 9] gives
a general introduction into the concepts mentioned here.
Let G be a T -group. Then G is a poly-C1 group and there is a series of normal
subgroups
G = G1 G2 Gm+1 = f1g
of G such that Gi =Gi+1 is innite cyclic and central for 1 i m (see 5]). Now
let ui 2 Gi be such that uiGi+1 generates Gi=Gi+1 : By induction, it follows that
each g 2 G can be written as a unique normal word g = ue11 : : : uemm with ei 2 Z:
We call the sequence (u1 : : : um) a poly-C1 generating sequence for G. Since
the normal series is central, uj ui] 2 Gj+1 and uj ui] is a normal word wij in
uj+1 : : : um: Relative to a poly-C1 generating sequence (u1 : : : um) the group
G has a presentation of the form
G = h u1 : : : um j uj ui = uiuj wij for 1 i < j m i
(1)
A presentation of the form (1) is called a polycyclic presentation of G. Using the
polycyclic presentation any word in (u1 : : : um) can be rewritten as a normal
word. There are algorithms for rewriting a group element to a word in normal
form, called collection algorithms (see 10], 6], 7], 11]). If every element g 2 G
has a unique normal form, then the polycyclic presentation (1) is called consistent. In the sequel we assume that every T -group is given by a consistent
polycyclic presentation of the form (1).
If a T -group is given by a consistent polycyclic presentation, then the multiplication of two elements in normal form can be performed by collecting the
concatenation of the two words to normal forms. Clearly, the exponents in the
normal form of the product are a function of the exponents in the factors.
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 3
Theorem 1 (P. Hall) Let G be a T -group and u1 : : : um a poly-C1 generat-
ing sequence for G: Let g = u1 1 : : : umm and h = u11 : : :umm be elements of G and
denote the sequence of exponents of g and h by and respectively.
Then there are polynomials fi 2 Q x1 : : : xm y1 : : : ym] such that
gh = uf11() ufmm():
Furthermore, each fi has the form
fi(x y) = xi + yi + pi(x1 : : : xi;1 y1 : : : yi;1):
Proof: The rst part is Theorem 6.5 of Hall 5]. The second part follows from
the fact that Gi =Gi+1 is central and from the observation, that the exponent of
ui in the product can be computed in the factor group G=Gi+1 : Therefore, only
the exponents of u1 : : : ui contribute to the exponent of ui in the product. 2
Lemma 2 Let G be a T -group with poly-C1 generating sequence u1 : : : um
and : G ! Ur (Z) a representation of G.
Then there are polynomials pij 2 Q x1 : : : xm] such that the (i j )-th entry of
the matrix
(u1 1 umm )
is equal to pij (1 : : : m ).
Proof: We have that (ui) = 1 + Mi where Mi is strictly upper triangular.
Therefore, Mir = 0 and
(ui i ) = (1 + Mi)i
=
1 X
i
k=0
k
Mik
=
;
Now the binomial coecient i is the polynomial
r;1 X
i
k=0
k
k Mi :
k
i(i ; 1) : : : (i ; k + 1)
k!
in i . Since the number of terms in the sum is independent of i each entry in
the matrix (ui i ) is a polynomial in i. It follows that the entries of the matrix
(u1 1 umm ) are polynomials in 1 : : : m:
2
3. Constructing extensions
Now let G be a T -group with polycyclic generating sequence u1 : : : um. We
want to construct an injective homomorphism : G ! Ur (Z) for some r > 0.
The method for constructing such a homomorphism described in this paper starts
with a representation m of the subgroup generated by um. For example we can
set
1 1
m(um) = 0 1 :
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 4
The main part of the algorithm consists of a method for constructing a representation k for the group generated by uk : : : um from a representation k+1
of the group generated by uk+1 : : : um. Starting with m as given above and
iterating this process m ; 1 times gives a representation for G of the required
type.
Suppose that H is a T -group, N a normal subgroup of H such that H=N
is an innite cyclic group and : N ! Ur (Z) a representation of N: Let h0
be an element of H such that hh0N i = H=N and let n1 : : : nt be a poly-C1
generating sequence of N:
The group algebra of N over the integers is denoted by ZN and
(ZN ) = f f : ZN ! Z j f is linearg
is its dual space. Let N act on ZN via multiplication from the right and dene
ah0 = h;1
0 ah0 for a 2 ZN: It is elementary to check that this denition can be
extended to a right action of H on ZN (compare Segal 9, Proposition 5.1]).
This induces an action from the left of H on (ZN ) as follows
(h f )(a) = f (ah) for f 2 (ZN ) h 2 H a 2 ZN:
For 1 i j r dene cij 2 (ZN ) to be the function that maps each a 2 ZN
to the (i j )-th entry of (a): The linear function cij is called a coecient of
the representation. By Lemma 2, there is a polynomial pij (x1 : : : xt) such that
cij (n1 1 : : : nt t ) = pij (1 : : : t): Let C be the Z-module generated by fcij j 1 i j rg: We call C the coecient space of .
Lemma 3 The coecient space C is an ZN -module. If is faithful, then C
is faithful.
Proof: For n 2 N the equation
(n cij )(a) = cij (an) =
r
X
cik (a)ckj (n) for all a 2 ZN
k=1
P
r
shows that n cij = k=1 ckj (n)cik 2 C for 1 i j r and that C is an
ZN -module.
For n 2 N suppose that n cij = cij for all 1 i j r: Evaluating cij at the
identity element of N gives
cij (1) = (n cij )(1) = cij (n) for 1 i j r
and (n) = (1): If is faithful, this implies n = 1 and shows that C is a faithful
N -module.
2
Now let S be the ZH -submodule of (ZN ) generated by C: As a Z-module, S
is generated by the the set fhk0 cij j 1 i j r k 2 Zg: This follows directly from
the fact that N is normal in H and that C is generated by fcij j 1 i j rg
as a Z-module.
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 5
Lemma 4 Let f 2 S: Then there is pf 2 Qx1 : : : xt] such that
f (n1 nt t ) = pf (1 : : : t):
Proof: For f 2 C this is clear by the remarks made earlier. By Theorem 1 there
are polynomials q1 : : : qt 2 Qx1 : : : xt] such that for k 2 Z
n1 nt t hk0 = hk0 n1 n2 +q ( ) nt t+qt( :::t; )
1
1
1
2
2
1
1
1
and each qi is a polynomial in x1 : : : xi;1 only. Setting i = qi(1 : : : i;1 ) we
get
(hk0 cij )(n1 1 nt t ) = cij (h;0 k n1 1 nt t hk0 ) = cij (n1 1+1 nt t+t )
= pij (1 + 1 : : : t + t):
Hence the polynomial corresponding to hk0 cij is pij (x1 + q1 : : : xt + qt):
2
Theorem 5 As a Z-module, S is nite-dimensional. Furthermore, there exists
a basis of S such that the corresponding matrix representation maps each h 2 H
to an upper triangular matrix with all diagonal entries 1:
Proof: Let R be the polynomial ring Q x1 : : : xt ] and let < be the reverse lexicographic order on the monomials of R dened as follows: xk11 xkt t < xl11 xltt
if there is an index 1 i t such that kt = lt : : : ki+1 = li+1 and ki < li. Dene the leading monomial lm(p) of an element p 2 R to be the largest monomial
with respect to < that occurs in p with non-zero coecient. Then < induces a
partial order on R dened by p < q if lm(p) < lm(q). It is routine to show that
< satises the descending chain condition and is translation invariant, i.e. that
p < q implies sp < sq for any s 2 R:
By the previous lemma, the partial order on R induces a partial order on S
by f < g if pf < pg . We will prove that for all h 2 H and f 2 S there is a
g 2 S with g < f such that h f = f + g:
By Theorem 1 there are polynomials qi 2 R in x1 : : : xi;1 such that for
h2H
h;1n1 1 nt t h = n1 1+1 nt t+t
where i = qi(1 : : : i;1): In particular qi < xi:
Let f 2 S and set f 0 = h f , then pf 0 = pf (x1 + q1 x2 + q2 : : : xt + qt):
By expanding and reordering the monomials in pf we get pf 0 = pf (x1 : : : xt) +
q(x1 : : : xt): The polynomial q is a sum of monomials from pf in which some
of the xi have been replaced by qi: The translation invariance of < and the fact
that qi < xi imply that q < pf : Since pf 0 and pf are polynomials corresponding
to elements of S there is an element g 2 S such that pg = q. This shows that
h f = f + g and g < f .
Now let f0 2 S and set fk+1 = h0 fk ; fk for k 0. Since the order < satises
the descending chain condition, there exists a K 0 such that h0 fK = fK .
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 6
Since S is generated as a Z-module by the set fhk0 cij j 1 i j r k 2 Zg we
have that S is nite-dimensional as a Z-module.
Let ff1 : : : fsg be a Z-basis of S. By subtracting basis elements from other
basis elements if necessary, we may assume that lm(pfi ) 6= lm(pfj ) for i 6= j .
Therefore, all elements of this basis are comparable with respect to the order <.
Suppose they have been ordered such that fi < fj if i < j . Then for h 2 H the
matrix of h with respect to this basis is the identity matrix plus a strictly upper
triangular matrix.
2
Proposition 6 Suppose that is a faithful representation of N . Then the representation : H ! GL(S) aorded by S is also faithful on N . Furthermore,
is a faithful representation of H or h0n = nh0 for all n 2 N .
Proof: The ZN -module S contains the ZN -submodule C: By Lemma 3, C is
faithful if is faithful.
Let h be an arbitrary element of H . Then h has the form hk0 n where n 2 N and
k 2 Z. Now suppose that (h) = 1: Then (hk0 n cij )(a) = cij (a) for all 1 i j r
or, equivalently, cij (h;0 k ahk0 n) = cij (a) for all 1 i j r and a 2 N . Since C is
a faithful ZN -module, this is equivalent to h;0 k ahk0 n = a for all a 2 N . Taking
a = 1 implies n = 1 and h;0 k ahk0 = a for all a 2 N , hence (hk0 ) = 1. Since
(h0) is unipotent by the previous theorem, we have that (h0) = 1, whence the
second statement.
2
4. The algorithm
We formulate the algorithm based on the results of the previous section. For
that we set
1 1
E2 = 0 1 :
Algorithm Representation
Input: a T -group G and a polycyclic generating sequence u1 : : : um of G.
Output: a faithful representation : G ! Ur (Z).
1. Let Hk be the subgroup of G generated by uk : : : um for k = 1 : : : m.
2. Let m : Hm ! U2(Z) be the representation of Hm given by m(um) = E2.
3. For i = m ; 1 m ; 2 : : : 1 do the following.
Let i+1 : Hi+1 ! Us(Z) be a faithful unipotent matrix representation
of Hi+1 :
If ui commutes with ui+1 : : : um then let i be the representation
0 (u )
i+1 j
i(uj ) = @
1
1A
1
0 1
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 7
for j = i + 1 : : : m and
01
B
B
i (ui) = B
B
B
@
...
1
C
C
C
:
C
C
1A
1
1
0 1
If ui does not commute with ui+1 : : : um then calculate the spaces
Ci+1 and Si+1 . Let i be the representation of Hi acting on Si+1 .
4. Return 1.
Proposition 7 The algorithm Representation returns a faithful representation of a T -group G by unipotent matrices over Z.
Proof: We prove that the representations i are faithful and by unipotent matrices. For m this is clear. Furthermore, suppose that this holds for i+1 . If ui
commutes with ui+1 : : : um then it is straightforward to see that the map i
is a group homomorphism, that it is faithful and by unipotent matrices. If on
the other hand ui does not commute with ui+1 : : : um, then i is by unipotent
matrices by Theorem 5. Furthermore, it is faithful by Proposition 6.
2
Corollary 8 Let G be a T -group, then G has a faithful nite dimensional representation by unipotent matrices over the integers.
Example 9 We consider the following group:
G = ha b c d e f j
b a] = c c a] = d d a] = e d b] = f e b] = f d c] = f ;1i
(trivial commutators of the generators have been omitted). The generators d e f
generate an Abelian subgroup. It follows that for rst two steps of the algorithm,
the rst half of Step 3 applies. Hence we get a representation 4 : hd e f i !
GL(6 Z) given by
01 p
B
0 1
B
B
4(dmen f p) = B
B
B
@
1 n
0 1
1
C
C
C
C
:
C
C
mA
1
0 1
Now we extend 4 to a representation of the group generated by c d e f . First
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 8
we calculate C4 . Since this is the space of all coecients, C4 is spanned by four
functions f1 f2 f3 f4, given by
f1(dm enf p) = 1
f2(dm enf p) = m
f3(dm enf p) = n
f4(dm enf p) = p:
In order to calculate S4 we let c act on these functions. For example:
c f4(dmen f p) = f4(c;1dm enf pc) = f4(dm enf p;m ) = p ; m:
Hence c f4 = f4 ; f2. In the same way one sees that c f1 = f1, c f2 = f2 and
c f3 = f3. Hence S4 = C4 . Using column convention, the representation 3 is
given by
01 m n p 1
B0 1 0 ;lC
3(cldmenf p ) = B
@0 0 1 0 C
A:
0 0 0 1
Analogously it is seen that C3 contains ve functions and S3 = C3 . The
representation 2 of the group generated by b c d e f is given by
01 l m n p 1
B
0 1 0 0 0 C
B
C:
k
l
m
n
p
B
2(b c d e f ) = B0 0 1 0 k ; lC
@0 0 0 1 k C
A
0 0 0 0 1
Hence C2 is spanned by six functions:
f1(bk cldm enf p) = 1
f2(bk cldm enf p) = k
f3(bk cldm enf p) = l
f4(bk cldm enf p) = m
f5(bk cldm enf p) = n
f6(bk cldm enf p) = p:
Before letting a act on C2 we calculate its conjugation action on the normal
subgroup generated by b c d e f . By an induction argument it is readily seen
that
l
a;1bk cldm enf pa = bk cl+k dm+l en+m f p;(2) :
Hence only the action of a on f6 promises to yield new functions. Now
l
k
l
m
n
p
k
l
+k m+l n+m p;(2l )
a f6(b c d e f ) = f6(b c d e f ) = p ; 2 :
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 9
So a f6 is not a linear combination of functions that we saw before. We dene
f7 by
f7(bk cldm enf p) = 12 l2 ; 12 l:
Going on:
l
a f7(bk cldm enf p) = f7(bk cl+k dm+l en+m f p;(2))
= 12 (l + k)2 ; 12 (l + k)
= 12 l2 ; 12 l + kl + 12 k2 ; 12 k:
Again we get a new function f8 dened by f8(bk cldmenf p ) = kl + 21 k2 ; 12 k.
Continuing:
l
a f8(bk cldm enf p) = f8(bk cl+k dm+l en+m f p;(2))
= k (l + k ) + 1 k 2 ; 1 k
2
2
1
1
2
= kl + 2 k ; 2 k + k2:
This leads yet again to a new function: f9(bk cldm enf p) = k2. But now the
process stops we have a f9 = f9. So we arrive at a 9-dimensional representation
we leave it to the reader to write down the matrices.
5. Implementation
We have implemented the algorithm in the computer algebra system GAP 4. In
this system the basic functionality for dealing with T -groups (e.g., representation
of words in the group and the collection algorithm) is already present. In this
section we outline the implementation of the algorithm Representation. For
this we revert to the language of Section 3.
First of all we note that if h0 commutes with the generators n1 : : : nt, then
the representation is extended to H without problems. So suppose that this is not
the case, and we have to compute the module S. The main problem that we have
to deal with is due to the fact that the dual space (ZN ) is innite-dimensional.
However since we are only interested in the nite-dimensional subspace S there
is a way around this problem. First we deal with the problem of representing
a function in S on a computer. If c is a coecient of then this is easy: we
only need to store the position ij such that c(n) is the ij -th coecient of the
matrix (n). Furthermore, for f = hk0 c we store the integer k together with
the position ij . Then we can calculate the value of f (a) for all a 2 ZN . Also
because any function in S is a linear combination of functions of the form hk0 c
we can represent all elements of S.
Now we consider the problem of doing linear algebra inside S. For that we
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 10
need to represent any element as a vector (list of coecients). Then using Gaussian elimination we can nd linearly independent sets, calculate matrices of endomorphisms of S and so on. For this we choose a nite set A of elements of
the group N and we represent an element f of S as a vector (f (a))a2A. The set
A is called a discriminating set for S if for every f 2 S such that f 6= 0 there
is an a 2 A such that f (a) 6= 0. Since S is nite-dimensional, discriminating
sets for S exist, and a smallest discriminating set consists of dim S elements.
Unfortunately we only have the following rather crude method for selecting a
discriminating set. Initially we let A be the set of all elements of N of degree
bounded by some limit d 1. Here we choose d large enough to ensure that A is
a discriminating set for C. In particular we have that 1, along with the generators n1 : : : nt are in A. Using this discriminating set we calculate the closure of
C under the action of h0. If A happens to be a discriminating set for S, then
this will give us a basis of S. If A is not a discriminating set for S , then two
things could go wrong: the resulting representation might not be a group homomorphism, or or it might not be faithful. In the rst case we increase the bound
d, and start again. On the other hand, if the resulting representation is a group
homomorphism, then it is also faithful. To see this we use the notations from the
proof of Proposition 6. In this case we get that f (h;0 k ahk0 n) = f (a) for f 2 S and a 2 A, where S is the space that we computed. Since S contains C we see
that h;0 k ahk0 n = a for all a 2 A. As A contains 1 we have that n = 1. Since S is a quotient module of S we have that h0 acts by a unipotent matrix on S as
well. So again we get that h0 commutes with all elements of A, and in particular
with the generators ni. But this is excluded, and therefore the representation is
faithful.
6. Examples
Table 1 lists a number of experimental results obtained with the implementation
pf the algorithm in GAP 4. All computations were done on a Linux system with
a 600MHz Pentium III processor and 40MB of working memory for GAP. We
use the following naming conventions. First, Uk (Z) denotes as before the full
unitriangular group over Z. Furthermore, F (k n) is the free nilpotent group
with k generators and class n, and E (k n) is the largest, nilpotent k-generator
group satisfying the n-th Engel identity. An asterisk at the name of a group
indicates that we have taken the largest torsion free quotient of the named group.
This can be done by factoring out the torsion subgroup using the GAP 4 package
`Polycyclic' 2].
From Table 1 we see that the algorithm is ecient enough to be able to deal
with goups with rather large Hirsch length. However, the running times and
dimensions obtained appear to increase exponentially. Also the dimensions of
the modules found by the algorithm are generally much smaller than the ones
constructed in 8].
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 11
Description
2 (Z)
3 (Z)
4 (Z)
5 (Z)
6 (Z)
F( 2, 2 )
F( 2, 3 )
F( 2, 4 )
F( 2, 5 )
F( 3, 2 )
F( 3, 3 )
h
j
]
(2 3)
(3 3)
(4 2)
U
U
U
U
U
x y
E
E
E
y x x y x y y y
]i
Class Hirsch length Dimension Time
1
1
2
0
2
3
3
0
3
6
7
0.4
4
10
16 12.3
5
15
35 181.3
2
3
3
0
3
5
6
0.3
4
8
10
2.3
5
14
20 119.7
2
6
6
0.2
3
14
17 14.9
5
6
11
1.7
3
6
6
0.3
4
17
26 151.4
6
11
19 19.0
Table 1: Experimental results for the algorithm. The fourth column displays the dimension of
the module obtained by the algorithm. The fth column contains the running time in seconds.
References
1] N. Bourbaki. Groupes et Algebres de Lie, Chapitre I. Hermann, Paris, 1971.
2] B. Eick and W. Nickel. Polycyclic, 2000. A GAP package, see 3].
3] The GAP Group, Aachen, St Andrews.
GAP {
Groups, Algorithms, and Programming, Version 4.2, 2000.
(http://www-gap.dcs.st-and.ac.uk/~gap).
4] W. A. de Graaf. Lie Algebras: Theory and Algorithms, volume 56 of NorthHolland Mathematical Library. Elsevier Science, 2000.
5] P. Hall. Nilpotent groups. Notes of Lectures given at the Canadian Mathematical Congress, University of Alberta, 1957.
6] M. Hall, Jr. The Theory of Groups. Macmillan, New York, 1959.
7] C. R. Leedham-Green and L. H. Soicher. Collection from the left and other
strategies. J. Symbolic Comput., 9:665{675, 1990.
8] E. H. Lo and G. Ostheimer. A practical algorithm for nding matrix representations for polycyclic groups. J. Symbolic Comput., 28(3):339{360, 1999.
9] D. Segal. Polycyclic Groups. Cambridge University Press, 1994.
10] C. C. Sims. Computation with Finitely Presented Groups. Cambridge University Press, Cambridge, 1994.
W. A. de Graaf and W. Nickel: Representations of torsion-free nilpotent groups 12
11] M. R. Vaughan-Lee. Collection from the left. J. Symbolic Comput., 9:725{
733, 1990.