Computing generators of the unit group of an integral abelian group ring

0.19 MB·English

by  Paolo Faccin

COMPUTING GENERATORS OF THE UNIT GROUP OF AN
INTEGRAL ABELIAN GROUP RING
PAOLOFACCIN,WILLEM A.DEGRAAF,ANDWILHELMPLESKEN
Abstract. We describe an algorithm for obtaining generators of the unit
groupoftheintegral groupringZGofafiniteabeliangroupG. Weusedour
3 implementationinMagmaofthisalgorithmtocomputetheunitgroupsofZG
1 forGoforderupto110. Inparticularforthosecasesweobtainedtheindexof
0 thegroupofHoechsmannunitsinthefullunitgroup. Attheendofthepaper
2 we describe an algorithm for the more general problem of finding generators
n ofanarithmeticgroupcorrespondingtoadiagonalizablealgebraicgroup.
a
J

9

1. Introduction
]
A
Let G be a finite abelian group. For a ring R we let RG be the group ring over
R
R, consisting of sums a g, with a ∈ R. We take R = Z and consider the
Pg∈G g g
h. unit group
t (ZG)∗ ={u∈ZG|there is a v ∈ZG with vu=1}.
a
m Higman ([14]) showed that (ZG)∗ = ±G×F, where F is a free abelian group.
[ Moreover, Ayoub and Ayoub ([1]) have established that the rank of F is 1(|G|+
2
1+t −2l),wheret is thenumber ofelementsofGoforder2,andl isthe number
1 2 2
of cyclic subgroups of G.
v
0 Hoechsmann ([15]) described a construction of a set of generators of a finite-
7 index subgroup of (ZG)∗, called the group of constructable units. Regarding this
7 construction he wrote “Does this method ever yield all units if n = |G| is not a
1
prime power? The answer seems to be affirmative for all n < 74.” In [15] this
.
1 question is not dealt with any further. When n=74 it is known that the group of
0
constructable units is of index 3 in the full unit group (see [16]). So the question
3
remains whether the constructable units generate the full unit group if |G|<74.
1
: In this paper we develop algorithms for computing generators of the unit group
v (ZG)∗. Using our implementation of these algorithms in the computer algebra
i
X system Magma ([4]) we have computed the unit groups for all abelian groups of
r order ≤ 110. We found 12 groups G of order less than 74 whose unit group is not
a generatedbythe Hoechsmannunits(namely,thegroupsoforder40,48,60,63and
65).
In the next section we start by collecting some well-known facts and immediate
observations concerning lattices, groups, and associative algebras. Also, in the
secondhalf of the section (Section 2.4) we describe our approachto computing the
unitgroupofthemaximalorderinacyclotomicfield. Thisisachievedbycombining
aconstructionbyGreither([12])ofafinite-indexsubgroupoftheunitgroup,along
with a Magma programby Fieker for “saturating”a subgroupat a given prime p.
The latter algorithm and its implementation will be described elsewhere.
Section 3 contains the main algorithm of this paper, namely an algorithm for
computing the unit group of an order O in a toral algebra A. The main idea is to
split A in its simple ideals e A, where the e are orthogonalprimitive idempotents.
i i
Thee Aarenumberfieldswithorderse O. Soinordertocomputetheirunitgroups
i i
we can use the effective version of the Dirichlet unit theorem (cf. [5], [19]). The
1
2 PAOLOFACCIN,WILLEMA.DEGRAAF,ANDWILHELMPLESKEN
basic step of the algorithm is, given two orthogonal idempotents ǫ , ǫ , to obtain
1 2
the unit group of (ǫ +ǫ )O given the unit groups of ǫ O, i=1,2.
1 2 i
In Section 4 we describe our method for obtaining generators of unit groups of
integral abelian group rings. Its main ingredients are the construction of the unit
groups of cyclotomic fields, and the algorithm of Section 3. We comment on the
running times of the implementation of the algorithm in Magma, and we give a
table containingallabeliangroupsofordersupto 110,where the constructable(or
Hoechsmann-) units do not generate the full unit group. For all these groups we
give the index of the group of constructable units in the full unit group.
Finally in the last section we indicate an algorithm to obtain generators of the
arithmetic group corresponding to a connected diagonalizable algebraic group de-
fined over Q. Again the main ingredient is the algorithm of Section 3.
In our main algorithms and implementations we make essential use of Fieker’s
implementation in Magma of an algorithm by Ge ([10]) to obtain a basis of the
lattice
{(α ,...,α )∈Zn |uα1···uαn =1}
1 n 1 n
of multiplicative relations of given elements u ,...,u in a number field.
1 n
Acknowledgement: we are very grateful to Claus Fieker for giving us early
access to his Magma programs MultiplicativeGroup and Saturation, without which
this project could not have been carried out.
2. Preliminaries
2.1. Lattices. In this paper we use the term“lattice”for a finitely generatedsub-
group of Zm. A lattice Λ ⊂ Zm has a basis, that is a subset u ,...,u such that
1 r
every u ∈ Λ can uniquely be written as u = r α u , with α ∈ Z. The lattice
Pi=1 i i i
Λ⊂Zm is called pure if Zm/Λ is torsion-free. (See [6], §III.16.)
Let Λ ⊂ Zm be a lattice with basis u ,...,u . We form the r ×m-matrix B
1 r
with rows consisting of the coefficients of the u with respect to the standard basis
i
of Zm. By computing the the Smith normal form of B we can effectively compute
the homomorphism ψ : Zm → Zm/Λ (cf. [21], §8.3). Let T denote the torsion
submodule of Zm/Λ. Then ψ−1(T) is the smallest pure lattice containing Λ. So in
particular,the Smith normalform algorithm gives a method to compute a basis of
the lattice V ∩Zm, where V is a subspace of Qm. For example, we can compute
the intersection of lattices this way.
Asobservedin[6],§III.16,alatticeΛispureifandonlyifitisadirectsummand
ofZm. Sointhatcase,bycomputingaSmithnormalform,wecancomputeabasis
of Zm such that the first r basis elements form a basis of Λ.
Remark 2.1. Computing a basis of the lattice V ∩Zm, where V is a subspace of
Qm, is called saturation. There are methods known for this, based on computing
a Hermite normal form, that are more efficient than the approach outlined above.
Going into these matters would lead us too far from the subject of this paper.
2.2. Toral algebras. We say that an associative algebra A over Q is toral if it is
semisimple, abelian and has an identity element, which we will denote by e. For
example the group algebra QG of a finite abelian group G is toral.
By the Wedderburn structure theorem (cf. [18], §3.5) a toral algebra A is a
direct sum A = A ⊕···⊕A , where the A are ideals that are isomorphic (as
1 s i
associative algebras) to field extensions of Q. A nonzero element e ∈ A is said
0
to be an idempotent if e2 = e . Two idempotents e ,e are called orthogonal if
0 0 1 2
e e = 0. Furthermore, an idempotent is called primitive if it is not the sum of
1 2
orthogonal idempotents. Now the decomposition of A into a direct sum of simple
ideals corresponds to a decomposition of the identity element e ∈ A as a sum of
COMPUTING GENERATORS OF THE UNIT GROUP OF AN INTEGRAL ABELIAN GROUP RING3
primitive orthogonalidempotents, e=e +···+e . Here e is the identity element
1 s i
of A , and vice versa, A =e A. We remark that there are algorithms to compute
i i i
the e , given a basis of A (cf. [8], [9]).
i
When A = QG, with G a finite abelian group, there is a very efficient way to
compute the primitive idempotents. Let χ:G→C∗ be an irreducible character of
G. Then
1
eχ = Xχ(g−1)g
|G|
g∈G
is an idempotent in CG. Moreover, the e , as χ runs over all irreducible charac-
χ
ters, are primitive orthogonal idempotents with sum e (cf. [6], Theorem 33.8). In
particular, they form a basis of CG. Let m denote the exponent of G, then all
irreducible characters χ have values in the cyclotomic field Q(ζ ). So the Galois
m
group Gal(Q(ζ )/Q) ∼= (Z/mZ)∗ acts on the irreducible characters. Now we sum
m
the e , for χ in an orbit of Gal(Q(ζ )/Q), and obtain the primitive orthogonal
χ m
idempotents of QG.
A subset O of a toral algebra A, containing the identity of A, is said to be an
order (or, more precisely, a Z-order) if there is a basis a ,...,a of A such that
1 m
O =Za +···+Za and a a ∈O for 1≤i,j ≤m. For example, ZG is an order
1 m i j
in QG. The unit group of O is
O∗ ={a∈O| there is b∈O with ab=e}.
Weconsidertheproblemofobtainingabasisofthelatticeofmultiplicativerelations
L={(α ,...,α )∈Zr |aα1···aαr =e},
1 r 1 r
where a ,...,a are given elements of O∗. Note that e A are number fields. So
1 r i
using Ge’s algorithm ([10]) we can compute bases of the lattices
L ={(α ,...,α )∈Zr |(e a )α1···(e a )αr =e }.
j 1 r j 1 j r j
Moreover, L=∩ L so we can compute a basis of L (see Section 2.1).
j j
2.3. Standard generating sets. LetU beafinitely-generatedabeliangroup. We
say that a set of generators g ,...,g of U is standard if
1 r
(1) for 1≤i≤s the order of g is d ,
i i
(2) for s+1≤i≤r the order of g is infinite,
i
(3) d |d for i<s,
i i+1
andthere arenoother relations. Soa standardsetofgeneratorsimmediately gives
an isomorphism of U to Z/d Z⊕···⊕Z/d Z⊕Zr−s.
1 s
Givingfinitely-generatedabeliangroupsbystandardgeneratingsetsyieldsstraight-
forwardalgorithmsforseveralcomputationaltasksconcerningthesegroups,suchas
computingtheindex ofasubgroup,andcomputingthe kernelofahomomorphism.
If an abelian group is given by a non-standard set of generators, then we can
compute a standardone by computing the lattice of all relations of the generators,
followed by a Smith normal form computation (cf. [21], §8.3). So, using the algo-
rithmindicatedintheprevioussection,wecancomputeastandardsetofgenerators
for a finitely-generated subgroup of O∗, where O is an order in a toral algebra.
Remark2.2. Formanycomputationalproblemsregardingfinitely-generatedabelian
groups it suffices to compute a Hermite normal form of the relation lattice. How-
ever, in our applications the main computational problem is to obtain the relation
lattice (see Table 4), the subsequent computation of the Smith normal form does
notbear heavilyonthe running time. Therefore,forour purposes,a Smith normal
form is the most convenient.
4 PAOLOFACCIN,WILLEMA.DEGRAAF,ANDWILHELMPLESKEN
2.4. Units of cyclotomic fields. Let n be a positive integer. We consider the
cyclotomic field Q(ζ ), where ζ is a primitive n-th root of unity. The ring of
n n
integers of this field is Z[ζ ]. By Dirichlet’s unit theorem the unit group Z[ζ ]∗
n n
is equal to T ×F, where F is a free abelian group of rank 1ϕ(n)−1, and T is
2
the group of roots of unity in Q(ζ ). The problem considered in this section is
n
to obtain generators of the unit group, Z[ζ ]∗. There are algorithms for this that
n
workfor any number field (cf. [5], [19]); but their complexity is suchthat it is only
practical to use them for n up to about 20 (depending on the hardware one uses,
of course). For this reason we sketch a different approach, using several results
from the literature. The situation is straightforwardwhen n is a prime power, see
Section 2.4.1. Then in Section 2.4.2 we describe what can be done when n is not
a prime power. Using these methods we obtained a list of the generators of the
unit groups Z[ζ ]∗, for n < 130. However, for several n the correctness of this list
n
depends on the Generalised Riemann Hypothesis, that is for n prime between 67
and 127, and for n=115,119,121,123,125,129(so 19 cases in total).
Throughoutwe setQ(ζ )+ =R∩Q(ζ ); then Q(ζ )+ =Q(ζ +ζ−1). By h+ we
n n n n n n
denote the class number of Q(ζ )+.
n
2.4.1. When n is a prime power. Suppose that n = pm is a prime power. For
1<a< n with gcd(a,p)=1 set
2
1−a1−ζa
ξ =ζ 2 n.
a n
1−ζ
n
Then ξ lies in the unit group of Q(ζ )+. Let U be the group generated by −1,
a n n
ζ and all ξ . Then for the index we have [Z[ζ ]∗ : U ] = h+ (this is obtained
n a n n n
by combining Corollary 4.13, Lemma 8.1 and Theorem 8.2 in [22]). It is known
that h+ = 1 if ϕ(n) < 66, and assuming the Generalised Riemann Hypothesis, we
n
have h+ = 1 when ϕ(n) < 162 (see the Appendix in [22]). So for those n we have
n
generators of the unit group.
2.4.2. When n is not a prime power. Here the situation is more difficult. First of
allweassumethatn6=2mod4,asforn=2mod4wehavethatQ(ζn)andQ(ζn)
2
are isomorphic. By E+ denote the unit group of Q(ζ )+. We use a finite-index
n
subgroupofE+definedbyGreither([12]). Herewebrieflydescribehisconstruction.
Let G = Gal(Q(ζ )/Q), and write the elements of G as σ , where gcd(a,n) = 1
n a
and σ (ζ )=ζa. For α= m σ ∈ZG and x∈Q(ζ ) define
a n n Pa a a n
xα =Yσa(x)ma.
a
Let n = s pei be the factorisation of n in distinct prime powers. Set S =
Qi=1 i
{1,...,s} and P ={I ⊂S |I 6=S}. For I ∈P we set n = pei.
S S I Qi∈I i
We consider arbitrary maps β :S →ZG, which we extend to maps (denoted by
thesamesymbol)β :P →ZG byβ(∅)=1,β({i})=β(i),andβ(I∪J)=β(I)β(J)
S
ifI∩J =∅. Nowletz ∈Q(ζn). ForI ∈PS setzI =1−znI,andz(β)=QI∈SzIβ(I).
Set t = − n β(I) ∈ ZG. Then for a with 1 < a < n and gcd(a,n) = 1 we
PI∈S I 2
consider
σ (z(β)) (1−σ )t
ξ (β)=ζda a , where d = a .
a n z(β) a 2
1
(Note thatfor n odd, ζ2 lies in Z[ζ ], whereasfor n evenwe have a oddand hence
n n
1−a is an integer.)
2
Following Greither we describe a good choice for β. First a small piece of nota-
tion: if g is anelement oforderm ofa group,then we setN =1+g+···+gm−1,
g
which lies in the corresponding integral group ring. Now consider an i∈S. Let G
i
COMPUTING GENERATORS OF THE UNIT GROUP OF AN INTEGRAL ABELIAN GROUP RING5
denote the Galois group of Q(ζn/pei)+ over Q. This group contains the Frobenius
automorphism Fi (by definition: Fii(ζn/peii)=ζnpi/peii). This yields the element NFi
in ZG . Now we define β(i) to be a lift of N to ZG.
i Fi
Let C be the subgroupof E+ generated by −1 and the ξ (β). Greither proved
β a
that C does not depend on the choice for the lifts of the N , and that it is of
β Fi
index h+i in E+, where
n β
s
iβ =Yeigi−1fi2gi−1;
i=1
here e , f and g are respectively the ramification, inertial and decomposition
i i i
degree of p in Q(ζ )+ (so that e f g = 1ϕ(n)).
i n i i i 2
NowletU denotethegroupgeneratedbyζ andC . Thenusing[22],Corollary
n n β
4.13, we get that [Z[ζ ]∗ :U ]=2h+i .
n n n β
WeusedaprogramthatClausFiekerhaswritteninMagma,V2.17-2(see[2]for
someofthe background). This program,givena finite-index subgroupV ofZ[ζ ]∗,
n
and a prime p, computes a subgroup V of Z[ζ ]∗, containing V, and such that the
n
index[Z[ζ ]∗ :V]isnotdivisiblebyp.eWeusedthisstartingwiththegroupU . For
n n
n < 130 and ϕe(n) ≤ 72 we have h+ = 1, and assuming the Generalised Riemann
n
Hypothesis, we have h+ = 1 for all n < 130 ([22], Appendix). Therefore we can
n
compute [Z[ζ ]∗ : U ], and get all primes dividing it. So in the end we arrived at
n n
the full unit group Z[ζ ]∗ for all n<130.
n
3. Unit groups of orders in toral matrix algebras
Let A be a toral algebra with identity e, and O ⊂ A an order. In this section
we consider the problem of computing a set of generatorsof O∗. First we consider
some special cases.
3.1. A simple toral algebra. Let A be a simple toralalgebra. Inother words,it
is isomorphic to a finite extension of Q. Now there are algorithms for computing
generators of an order in a number field (the effective version of the Dirichlet unit
theorem, see [5], [19]). We use these to compute generators of O∗ as well.
3.2. Two idempotents. Let A be a toral algebra with identity e, and e ,e ∈A
1 2
orthogonal (but not necessarily primitive) idempotents with e + e = e. Set
1 2
A = e A, then A = A ⊕A . Let O be an order in A, then O = e O is an order
i i 1 2 i i
in A . Here we suppose that we havegeneratorsof O∗, i=1,2,andthe problemis
i i
to find generators of O∗.
Set J = (e O∩O)+(e O∩O). Since this is an ideal in O we can form the
1 2
quotient R = O/J. Consider the maps ϕ : e O → R defined by ϕ (e a) = a+J,
i i i i
for a ∈ O. (Note that this is well-defined: if e a = e b then a−b = e (a−b)
1 1 2
so it lies in J.) These are surjective ring homomorphisms with respective kernels
e O∩O.
i
Lemma 3.1. O ={a +a |a ∈e O and ϕ (a )=ϕ (a )}.
1 2 i i 1 1 2 2
Proof. Let a ∈ O and set a = e a then a = a +a and a ∈ e O. Moreover
i i 1 2 i i
ϕ (a ) = a+J = ϕ (a ). Conversely, let a,b ∈ O be such that ϕ (a ) = ϕ (a ),
1 1 2 2 1 1 2 2
where a = e a and a = e b. Then a−b ∈ J, whence a = b+u +u where
1 1 2 2 1 2
u ∈e O∩O. Thereforee a=e b+u sothata +a =e a+e b=e b+e b+u =
i i 1 1 1 1 2 1 2 1 2 1
b+u which lies in O. (cid:3)
1
Corollary 3.2. O∗ ={a +a |a ∈(e O)∗ and ϕ (a )=ϕ (a )}.
1 2 i i 1 1 2 2
So in order to compute generators of O∗ we perform the following steps:
(1) Compute bases of O =e O.
i i
6 PAOLOFACCIN,WILLEMA.DEGRAAF,ANDWILHELMPLESKEN
(2) Compute bases of O ∩O, of J =(O ∩O)+(O ∩O), and set R=O/J.
i 1 2
(3) Compute generators of the groups H =ϕ (O∗)⊂R∗ and H =H ∩H .
i i i 1 2
(4) Compute generators of the groups M =ϕ−1(H)⊂O∗.
i i i
(5) Compute generators of the group O∗ = {a +a | a ∈ M and ϕ (a ) =
1 2 i i 1 1
ϕ (a )}.
2 2
3.2.1. Implementation. We comment on the implementation of the steps of the al-
gorithm. Step(1)isdonebyaHermitenormalformcomputation. Theintersections
in Step (2) are computed using the techniques indicated in Section 2.1. Note that
a basis of J is obtained by concatenating the bases of O ∩O. The ring R =O/J
i
can be constructed by a Smith normal form computation.
ForStep(3)weassumethate Aisisomorphictoanumberfield(inotherwords,
2
that e is a primitive idempotent). When using the algorithm, this can always be
2
arranged (see Section 3.3). Then e O an order in it. We set I =e O∩O and use
2 2
the isomorphism R ∼= (e O)/I. Subsequently we use algorithms described in [13],
2
[17] to compute a standard generating set of (e O/I)∗. So computing H , H as
2 1 2
subgroups of (e O/I)∗ we can perform the operations of Step (3).
2
In Step (4) we view ϕ as a homomorphism O∗ →H . We use this to compute
i i i
generatorsofthekernelofϕ aswellaspre-imagesofthegeneratorsofH. Together
i
thesegeneratethegroupM . AsremarkedinSection2.3,wecancomputestandard
i
generating sets of the groups O∗. Using these, it is straightforward to obtain a
i
standard generating set for the subgroups M . Then we restrict ϕ to obtain a
i i
homomorphism ϕ :M →H.
i i
Now we come to Step (5). Let h ,...,h , a ,...,a , b ,...,b be standard
1 r 1 s 1 t
generating sets of H, M and M respectively. Set
1 2
Λ={(α ,...,α ,β ,...,β )∈Zs+t |ϕ (aα1···aαs)=ϕ (bβ1···βet)}.
1 s 1 t 1 1 s 2 1 t
Then
O∗ ={aα1···aαs +bβ1···bβt |(α ,...,α ,β ,...,β )∈Λ}.
1 s 1 t 1 s 1 t
Moreover, Λ is a lattice, hence has a finite basis. Furthermore, the elements of
O∗ corresponding to the elements of a basis of Λ generate O∗. So the problem of
finding a generating set of O∗ is reduced to finding a basis of Λ.
Define µ ,ν ∈Z by
ij ij
r
ϕ1(ai)= Yhµjij
j=1
r
ϕ2(bi)= Yhνjij.
j=1
A small calculation shows that (α ,...,α ,β ,...,β )∈Λ if and only if
1 s 1 t
s t
(1) Xµijαi−Xνkjβk =0modord(hj)
i=1 k=1
for 1 ≤ j ≤ r (and where ord(h ) denotes the order of h ). Let S be the integral
j j
matrix with columns
(µ ,...,µ ,−ν ,...,−ν ,ord(h ))
1j sj 1j tj j
for 1 ≤ j ≤ r. We compute a basis the integral kernel of S, which is the set of all
v ∈Zs+t+1 suchthatvS =0. For eachv inthis basiswe takethe vectorconsisting
of the first s+t coordinates. This way we obtain a basis of Λ.
COMPUTING GENERATORS OF THE UNIT GROUP OF AN INTEGRAL ABELIAN GROUP RING7
3.3. The general case. Now let A be a toral algebra, e ,...,e its primitive
1 m
orthogonalidempotents withsume,andA =e Athecorrespondingsimpleideals.
i i
Let O be an order in A; then e O is an order in A , and as indicated in Section
i i
3.1, we can compute generators of the unit groups (e O)∗. Then for j = 2,3,...
i
we set ǫ = e +···+e and we apply the algorithm of Section 3.2 to the algebra
j 1 j
ǫ A with its order ǫ O, and two idempotents ǫ and e , yielding the unit group
j j j−1 j
(ǫ O)∗. When the algorithm terminates we have the unit group O∗.
j
4. Units of integral abelian group rings
Let G be an abelian group. As seen in Section 2.2, it is straightforward to
compute primitive orthogonal idempotents e ,...,e ∈ QG such that e (QG) is
1 r i
isomorphictoafieldextensionofQ. LetmdenotetheexponentofG,then(cf. [1])
td
QG∼=MMQ(ζd),
d|m i=1
where Q(ζ ) is the cyclotomic field of order d, and t is the number of cyclic
d d
subgroups of G of order d. In particular, e (QG)∼=Q(ζ ). We consider the order
i di
ZG in QG. We have that e (ZG) is isomorphic to Z[ζ ], and by the results of
i di
Section 2.4, we have generators of Z[ζ ]∗ for d < 130. So for small groups G we
di i
canapply the algorithmof Section 3 to obtaingeneratorsof the unit group(ZG)∗.
Using our implementation of the algorithms in Magma, we have carried this out
for all abelian groups of orders up to 110. In Table 4 we collect some timings and
other data related to the algorithm.
G ϕ(exp(G)) |digits| t t
m tot
C 24 7.4 314 328
70
C 32 63.4 1131 1183
80
C 24 159.5 1043 1078
90
C 72 3.7 2352 2446
91
C 32 31.2 2322 2373
96
C ×C 16 181.3 1575 1781
2 48
C ×C ×C 8 54.6 1031 1267
2 2 24
C ×C ×C 4 22.5 537 725
2 4 12
C 40 217.6 3822 3942
100
C ×C 20 66352.6 414569 426654
2 50
C ×C 8 379.7 1227 1425
5 20
C ×C 4 275.2 1131 1332
10 10
Table 1. Runtimes (in seconds) for the algorithm to compute
generators of (ZG)∗. The first column lists the isomorphism type
of G, and the second the value of the Euler ϕ-function on the
exponentofG. Thethirdcolumnhastheaveragenumberofdigits
ofthecoefficientsoftheunitsoutputbythealgorithm(withrespect
tothestandardbasisofZG). Thefourthcolumndisplaysthetime
spenttocomputemultiplicativerelationsincyclotomicfields. The
last column has the total time spent by the algorithm.
From Table 4 we see that the running time is dominated by the time needed to
compute multiplicative relations in cyclotomic fields (Ge’s algorithm). This algo-
rithmneedstoworkharderifthedegreesofthefieldsthatoccurarehigher. Indeed,
the running times generally increase when ϕ(exp(G)) increases (note that this is
the highest degree of a cyclotomic field occurring in the decomposition of QG).
8 PAOLOFACCIN,WILLEMA.DEGRAAF,ANDWILHELMPLESKEN
However, also the size of the elements of which we need to compute multiplicative
relations plays a role. For some groups the average number of digits of a unit, as
outputby the algorithm,is veryhigh. This is seenmostdramaticallyfor C ×C .
2 50
Notealsothatthesize(i.e.,theaveragenumberofdigitsoftheircoefficients)ofthe
units output by the algorithmis far frombeing optimal; indeed, for G=C ×C ,
2 50
the unit group (ZG)∗ is also generated by the Hoechsmann units (see below).
Various constructions of finite index subgroups of (ZG)∗ have appeared in the
literature (see [20] for an overview). Among these a construction by Hoechsmann
([15])seemstoyieldsubgroupsofparticularlysmallindex. Asanapplicationofour
algorithm, we have compared the full unit groups with the groups of Hoechsmann
units, for G of size up to 110.
We brieflydescribeHoechsmann’sconstruction. First,letC be acyclicgroupof
order n, generated by the element x. For i≥0 and y ∈C we set
s (y)=1+y+···+yi−1.
i
Let i,j be integers with 0 < i,j < n and gcd(i,n) = gcd(j,n) = 1. Let k,l be
positive integers with li=1+kn. Set
u (x)=s (xi)s (xj)−ks (x).
i,j l i n
Then u (x) is a unit in (ZC)∗. Let Θ(C) denote the set of all units constructed
i,j
in this way. We let H be the groupofunits in (ZG)∗ generatedby allΘ(C), where
C ranges over the cyclic subgroups of G of order > 2, along with ±G. It is called
the group of constructable units of ZG. By [15], Theorem 2.5, H is a subgroup of
finite index of (ZG)∗.
Ingeneralthe set consisting ofthe u (x) is a heavily redundantgeneratingset.
i,j
We note, however, that [16] Theorem 1.1, gives a non-redundant generating set of
H, if the group H =(Z/mZ)∗/{±1}is cyclic, where m is the exponent of G. We
m
remark that for small m the group H is cyclic with only few exceptions. The
m
values of m up to 120,for whichH is not cyclic, are24,40,48,56,60,63, 65,72,
m
80, 84, 85, 88, 91, 96, 104, 105, 112, 117, 120.
WeletHind(G)betheindexofHin(ZG)∗,andwecallthisnumbertheHoechs-
mann index. We have computed the Hoechsmann indices for all abelian groups of
orders up to 110. For most groups the index is 1. The groups for which it is not 1
are listed in Table 4, along with the corresponding Hoechsmann indices.
From Table 4 we see that on many occasions when Hind(G) 6= 1 we have that
|G| = m, with H not cyclic. Also, for the groups considered, if for one group G
m
of order m we have Hind(G) 6=1, then the same holds for all groups of that order
(except when (ZG)∗ has rank 0).
Remark4.1. ForthegroupsC ,withpaprimebetween67and120thecorrectness
p
of our computation depends on the Generalised Riemann Hypothesis (see Section
2.4).
5. Computing generators of arithmetic groups of diagonalisable
algebraic groups
In this section we use the term D-group as short for for diagonalisable algebraic
subgroupofGL(n,C), defined overQ. We considerthe problemto compute gener-
ators of the arithmetic group G(Z)=G∩GL(n,Z) for a given connected D-group
G. We assume that the group G is given by its Lie algebra g ⊂ gl(n,C) which in
turnisgivenbyabasisofmatriceswithcoefficientsinQ. Inthesequelsuchabasis
will be called a Q-basis.
We have two subsections: the first is devoted to characters, and in the second
we outline our algorithm.
COMPUTING GENERATORS OF THE UNIT GROUP OF AN INTEGRAL ABELIAN GROUP RING9
G Hind(G) G Hind(G)
C 2 C 2
40 84
C ×C 2 C ×C 2
2 20 2 42
C ×C ×C 2 C 2
2 2 10 85
C 2 C 3
48 90
C ×C 2 C ×C 3
2 24 3 30
C ×C ×C 2 C 3
2 2 12 91
C ×C 4 C 8
4 12 96
C 2 C ×C 16
60 2 48
C ×C 2 C ×C ×C 32
2 30 2 2 24
C 3 C ×C ×C ×C 16
63 2 2 2 12
C ×C 3 C ×C ×C 64
3 21 2 4 12
C 2 C ×C 64
65 4 24
C 3 C 7
74 98
C 4 C ×C 343
80 7 14
C ×C 8 C 2
2 40 104
C ×C ×C 16 C ×C 2
2 2 20 2 52
C ×C ×C ×C 32 C ×C ×C 2
2 2 2 10 2 2 26
C ×C 8 C 4
4 20 105
Table 2. Hoechsmann indices for abelian groups of orders up to
110, for which this index is not 1.
5.1. Computing the characters. We recall that a character of a group H is a
homomorphismχ:H →F∗, whereF∗ denotesthe multiplicativegroupofnonzero
elements of a field F. In this section we let G ⊂ H ⊂ GL (C) be two connected
m
D-groups defined over Q. A well-known theorem (cf. [3], Proposition 8.2) states
that there are characters χ ,...,χ : H → C∗ such that G is the intersection of
1 t
their kernels, i.e.,
G={h∈H |χ (h)=1 for 1≤i≤s}.
i
TheproblemconsideredinthissectionistofindanumberfieldK,andcharacters
χ ,...,χ : H → K∗ such that G is the intersection of their kernels. We require
1 t
that K and the χ be explicitly given, i.e., that we know the minimum polynomial
i
ofa primitive elementofK,andthatfor a givenh∈H we caneffectively compute
χ (h). We assume that we are given the Lie algebras h,g of H and G respectively,
i
in terms of Q-bases.
Proposition5.1. LetX ∈GL (C)besuchthatH =XHX−1consistsofdiagonal
m
matrices. Set h˜=XhX−1, ˜g=XgX−1. Then h˜,e˜g consist of diagonal matrices as
well. Set
Λ(h˜)={(e ,...,e )∈Zm |e α +···+e α =0 for all diag(α ,...,α )∈h˜},
1 m 1 1 m m 1 m
and similarly define Λ(˜g). Then Λ(h˜) ⊂ Λ(˜g) ⊂ Zm are pure lattices (cf. Section
2.1). Let e = (e ,...,e ) ∈ Zm, for 1 ≤ i ≤ s, be a basis of Λ(˜g) such that
i i,1 i,m
e ,...,e is a basis of Λ(h˜). For 1 ≤ i ≤ t define χ : H → C∗ by χ (h) =
t+1 s i i
αei,1···αei,m, where XhX−1 = diag(α ,...,α ). Then G, as subgroup of H, is
1 m 1 m
the intersection of the kernels of the χ .
i
Proof. Note that h˜ is the Lie algebra of H. Therefore, it consists of diagonal
matrices. Since ˜g ⊂ h˜ we get the same propeerty for ˜g. The lattices Λ(h˜), Λ(˜g) are
obviously pure. So as noted in Section 2.1, the required basis of Λ(˜g) exists.
A character χ of H is given by a sequence (e ,...,e ) of integers such that
1 m
χ(diag(α1,...,αem)) =eαe11···αemm. This follows from the same statement for the
e
10 PAOLOFACCIN,WILLEMA.DEGRAAF,ANDWILHELMPLESKEN
full diagonal group D , along with the fact that a character of H extends to
m
a character of D ([3], Proposition 8.2). Furthermore, the differeential of χ is
m
given by dχ(diag(α1,...,αm)) = e1α1 + ···+ emαm. So since h˜ consists ofeall
diag(α ,...e,α ) with e α +···+e α = 0 for all (e ,...,e ) ∈ Λ(h˜) we get
1 m 1 1 m m 1 m
that
H ={diag(α ,...,α )|αe1···αem =1 for all (e ,...,e )∈Λ(h˜)}.
1 m 1 m 1 m
e
Indeed, the latter is a connected algebraic group with the same Lie algebra as H.
An analogousstatement holds for G=XGX−1 and ˜g. This implies the statement
of the proposition. e (cid:3)
So in order to solve the problem of this section we need to say how the various
objects in Proposition 5.1 can be computed.
Let B ⊂ M (Q) be the associative algebra with one generated by the basis
m
elements of h. Then, using the terminology of Section 2.2, B is toral (indeed: its
elements can simultaneously be diagonalized). Now using algorithms given in for
example [7], [11] we can compute a splitting element b ∈ B. This means that
0
b generates B (as associative algebra). By repeatedly factoring polynomials over
0
number fields, we construct the splitting field K of the minimal polynomial of
b over Q. After factoring this polynomial over K we find the eigenvalues and
0
eigenvectors of b . From this we get a matrix X ∈ GL (K) such that Xb X−1 is
0 m 0
diagonal. WenotethatthisalsoimpliesthatXhX−1 consistsofdiagonalmatrices.
As this is the Lie algebra of H = XHX−1, we get the same for H, as the latter
group is connected. e e
Since we have given a basis of h we can compute a basis of XhX−1. Cor-
responding to a basis element diag(α ,...,α ) we construct the linear equation
1 m
α x +···α x =0 in the unknowns x . We can express the α as vectors in Qd,
1 1 m m i i
where d is the degree of K. So by solving these equations we get a basis of the
space
V(h˜)={(a ,...,a )∈Qm |a α +···+a α =0 for all diag(α ,...,α )∈h˜}.
1 m 1 1 m m 1 m
Now Λ(h˜)=V(h˜)∩Zm, so as seen in Section 2.1, we can compute a basis of Λ(h˜).
Furthermore,usingthemethodsoutlinedinthesamesection,wegetabasisofΛ(˜g)
that contains a basis of Λ(h˜). Then the characters χ are easily constructed as in
i
Proposition 5.1.
5.2. Putting the pieces together. Now let G ⊂ GL (C) be a connected D-
m
group, defined over Q. We assume that we have given its Lie algebra g ⊂ gl (C)
m
by a Q-basis. We describe an algorithm for obtaining a finite set of generators of
the group G(Z)=G∩GL (Z).
m
Lemma 5.2. Let A⊂M (C) be the associative algebra with one generated by the
m
basis elements of g. Let A∗ denote the set of invertible elements of A. Then A∗ is
a D-group defined over Q with Lie algebra A. Furthermore, G⊂A∗.
Proof. Note that A has a basis consisting of matrices with coefficients in Q. Let
A(Q) denote the Q-algebraspannedby sucha basis. Consider the linear equations
that define A(Q) as a subspace of M (Q). Then those equations define A∗ as
m
a subgroup of GL (C). Since the basis elements of g can be simultaneously be
m
diagonalised, the same holds for the elements of A∗. It follows that A∗ is a D-
group, defined over Q.
Since A∗, as subgroup of GL (C), is given by linear equations, its Lie algebra,
m
as subalgebra of gl (C), is given by the same equations. Hence the Lie algebra of
m
A∗ is A, where the Lie product is given by the commutator. So since g ⊂ A, we
also have G⊂A∗, as G is connected. (cid:3)