8 Intersections, centralizers, and normalizers
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LEMMA 8.51 If U is normalized by V, then
homomorphism ϕ:V→VU/U.
is the kernel of the natural
in the general case. This
Now, we shall introduce a method to compute
method refines the orbit-stabilizer approach of Lemma 8.50 and it uses the
special case of Lemma 8.51. The basic idea of the method is an induction
downwards along a normal series of G with elementary abelian factors. Let
N=Nl be the last nontrivial subgroup in such a series and suppose that N is
a p-group of rank d, say. By induction, we assume that the intersection in
the factor G/N is already computed; that is, we are given an induced polycyclic
sequence for
. Thus we can read off an
induced polycyclic sequence for
using the preimages method of
Section 8.4.
LEMMA 8.52 Let ϕ:U→UN/N be the natural homomorphism. Then
.
PROOF This follows directly from
Thus
, and
preimages of the subgroup
We note that
intersection.
LEMMA 8.53 Let
.
can be computed effectively as
using the methods of Section 8.4.2.
holds, and it remains to determine the latter
. Then:
a) M is normalized by K and thus K acts on N/M by conjugation.
b) Every k∈K can be written as k=vknk for vk∈V and nk∈N and the coset
nkM is uniquely defined by k.
c)
is a well-defined derivation.
PROOF a) First we note that
. Thus
M is normalized by V, since N is normal in G. Also, M is normalized by N,
since N is abelian. Thus M is normalized by VN and hence by K≤VN.
b) As K≤VN, we can write every element in K as product in VN. Clearly,
this factorization is unique modulo
.
c) By b) it follows that δ is a well-defined mapping. Clearly, we have that
δ(1)=M. Further, since
we have δ(kh)=δ(k)hδ(h).
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We note that the subgroup M of N can be computed by Lemma 8.51,
since N is a normal subgroup of G. We also note that N and thus also N/M
are elementary abelian p-groups and thus we can identify N/M with the
additive group of where e is the rank of N/M. Using this identification
and switching to additive notation, we can write the conjugation action of
and we consider δ as a
K on N/M as a homomorphism
derivation of the form
. This yields the following homomorphism
corresponding to the affine action (see Definition 8.43) of ϕ and δ on N/M:
Note that this affine action can be computed effectively by its definition.
It yields the following characterization of
.
LEMMA 8.54
where K acts via α on
.
PROOF We noted already that
By Lemma 8.50 we have that
where K acts by multiplication on the right on the cosets
L\G. We prove that StabK(L)=StabK(v) for
. First, let
k∈Stab K(L). Then k∈KʝL and thus k∈V. Hence nkM=M and δ(k)=0.
Therefore k∈StabK(v). Now, let k∈StabK(v). Then δ(k)=0 and thus nk∈M. As
, we have k∈L. Thus
.
Hence we obtain an effective method to compute the intersection
. Similar
to the method proposed by Lemma 8.50, this algorithm is also based on the
orbit-stabilizer algorithm of Section 8.6. But instead of computing one large
orbit of right cosets and its corresponding stabilizer, it determines several
smaller orbits of vectors and their corresponding stabilizers. This is usually
much more efficient.
8.8.2 Centralizers
We shall introduce a method of computing CG(g)={h∈G hg=gh} for an element
g∈G. We assume that g is given as a collected word in the generators of G
and we aim to determine an induced polycyclic sequence for CG(g).
The centralizer CG(g) is the stabilizer of g under the conjugation action
of G on its elements. Hence CG(g) can be computed using the orbit-stabilizer
algorithm of Section 8.6. However, the orbit of gG needs to be computed and
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stored explicitly in this approach. This can be time and space consuming.
We introduce a more effective approach in the following.
The basic idea of the method is an induction downwards along a normal
series with elementary abelian factors. Let N=Nl be the last nontrivial term
in such a series and suppose that N is a p-group of rank d, say. By induction,
we assume that the centralizer in the factor group G/N is already computed;
that is, we have given an induced polycyclic sequence for C/N=CG/N(gN).
So we can read off an induced polycyclic sequence for C. The following
lemma is elementary.
LEMMA 8.55 The mapping
derivation.
is a
Now we identify N with the additive group of and we switch to additive
notation. Then we can write the conjugation action of C on N as a
homomorphism
and we can consider δ as a derivation of
the form
. These can be combined to give an affine action:
This affine action yields the following characterization of CG(g).
LEMMA 8.56 CG(g)=StabC((0,…, 0, 1)) where C acts on
via α.
PROOF The stabilizer in C of (0,…, 0, 1) under the action of α is the kernel
of the derivation δ. By the definition of δ, we have ker(δ)=CC(g). As CG(g)≤C,
we have that CC(g)=CG(g) which completes the proof.
Hence we obtain an effective method to compute CG(g). Similar to the
method using a single orbit-stabilizer application, this algorithm is also
based on the orbit-stabilizer algorithm of Section 8.6. But this induction
method computes several relatively small orbits of vectors instead of one
relatively large orbit of elements in a polycyclic group. Thus the induction
method is usually more effective than the single orbit-stabilizer application.
8.8.3 Normalizers
For a given subgroup U≤G, we now introduce a method of computing its
normalizer NG(U):={h∈G|hU=Uh}. We assume that U is given by an induced
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polycyclic sequence and we aim to compute an induced polycyclic sequence
for NG(U).
The normalizer NG(U) is the stabilizer of U under the conjugation action
of G on its subgroups. Hence NG(U) can be computed using the orbit-stabilizer
algorithm of Section 8.6. However, the orbit UG needs to be computed and
stored explicitly in this approach. This can be time and space consuming.
Thus we introduce a refinement of this approach that is usually more efficient.
Once again, we use an induction downwards along a normal series with
elementary abelian factors. Let N=Nl be the last nontrivial term in such a
series such that N is an elementary abelian p-group of rank d, say. By
induction, we assume that the normalizer in the factor group G/N is already
computed; that is, we know an induced polycyclic sequence for R/N:=NG/N
(UN/N), and we can read off an induced polycyclic sequence for R. Now we
proceed in several steps.
As a first step we determine
using the idea of Lemma 8.51.
This is a usually very effective preliminary computation. As a second step,
we then compute S:=StabR(M). For this purpose we identify N with the
additive group of and we switch to additive notation. Then the conjugation
action of R on N translates to a homomorphism
and we
have to compute the stabilizer of the subspace of N corresponding to M
under the matrix action of R on N.
The following lemma investigates M and S further.
LEMMA 8.57 Let S:=StabR(M) with
. Then:
a)
M is normal in UN and thus UN≤S. Further, U/M is a complement
to N/M in UN/M.
b)
NG(U)=StabS(U/M) where S acts on the set of complements to N/M in
UN/M by conjugation.
PROOF a) M is normal in U, since N is normal in G, and M is normal in N,
since N is abelian. Thus M is normal in UN. As
, it follows that
U/M is a complement to N/M in (U/M)(N/M)=UN/M.
b) NG(U) normalizes UN and
, since N is normal in G. Thus
NG(U)≤S and hence we find that NG(U)=NS(U). As S normalizes M, we
have NS(U)=NS(U/M)=StabS(U/M), where S acts by conjugation on the set
of complements.
Hence it remains to determine the stabilizer of U/M under the conjugation
action of S on the complements to N/M in UN/M. This situation has already
been discussed in Section 8.7.1. There we showed in Lemma 8.44 that S
acts affinely on Z1(UN/N, N/M). Using this action, we can describe the Sconjugacy class of U/M in S/M as an orbit under this affine action; see
Theorem 8.45.
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Thus as a third step in our algorithm, we determine the elementary
abelian p-group Z1(UN/N, N/M) and the affine action of S on this group.
Then, we identify the elementary abelian p-group Z1(UN/N, N) with the
additive group of and we switch to additive notation. The affine action of
S translates into an action homomorphism
. Now it
remains to calculate a stabilizer of the trivial affine vector under this action
of S. This is summarized in the following lemma.
LEMMA 8.58 NG(U)=StabS((0,…, 0, 1)) where S acts on
action of Lemma 8.44.
via the affine
Hence we obtain an effective method to compute NG(U). The induction
downwards along the normal series with elementary abelian factors splits
the computation into a sequence of induction steps. In each induction step we
have to perform a variety of computations. In particular, each induction step
requires two applications of the orbit-stabilizer algorithm. These two
applications are usually the time and space consuming parts of the algorithm.
8.8.4 Conjugacy problems and conjugacy classes
There are various related problems to the determination of centralizers
and normalizers. For example, with g, h∈G and U, V≤G there are the
following problems.
(1) Determine the conjugacy class gG or UG explicitly.
(2) Check whether there exists x∈G with gx=h or Ux=V.
(3) Determine all conjugacy classes of elements or subgroups in G.
All these problems can be solved by minor modifications of the algorithms in
the Sections 8.8.2 and 8.8.3. We discuss these problems for the elements of G
here briefly and we show that they can be solved by variations of the centralizer
algorithm. The corresponding problems for subgroups can be solved by similar
variations of the normalizer algorithm.
Problem (1) can be solved directly by the centralizer algorithm. If an induced
polycyclic sequence for CG(g) is given, then a transversal T for CG(g) in G
can be read off from Lemma 8.33. This yields that gG={gt|t∈T}.
Problem (2) requires a variation of the centralizer algorithm. In each
induction step of this algorithm a stabilizer computation is performed. To
solve Problem (2) we also compute its underlying orbit and check whether
the element induced by h is contained in this orbit. If not, then g and h are
not conjugate. If so, then we modify h to its conjugate hy such that hy
induces the same element as g in the considered factor. Then we proceed
to the next induction step.
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Problem (3) also requires a variation of the centralizer algorithm. In
each induction step of this algorithm a stabilizer computation of a given
element is performed. To solve Problem (3) we have to compute the orbits
and the stabilizer of all possible elements and then use all these elements
and their orbits and stabilizer in the next induction steps.
On a historical note, computing centralizers and conjugacy classes in finite
p-groups using the downwards induction method with polycyclic presentations
was introduced by V.Felsch and J.Neubüser in [FN79]. They used an
implementation of the algorithm to find a counterexample of order 234 to the
‘class-breadth’ conjecture. The conjecture was that, for a finite p-group G,
c(G)≤b(G)+1, where c(G) is the nilpotency class and b(G) is the size of the
largest conjugacy class of G.
Exercises
1. Use the algorithm described above to compute the centralizer of (2, 3)
in G:=Sym(4), using the normal series with
.
2. Devise an alternative algorithm for computing the centralizer and
conjugacy class of an element g in a polycyclic group G, which works by
computing CGi(g) and gGi for i=n, n-1,…, 1, where G=G1≥G2≥…≥Gn+1=1
is a polycyclic sequence for G.
8.9 Automorphism groups
Let
be a polycyclic group defined by a refined consistent
polycyclic presentation. In this section we shall outline an algorithm to
compute the automorphism group Aut(G); that is, we want to determine
generators for Aut(G) and the order |Aut(G)|.
The method that is outlined here was first described by D.J.S.Robinson in
[Rob81] and an implementation has been investigated and outlined in detail
by M.J.Smith in [Smi94]. We note at this point that a dual approach can be
used to check isomorphism between two polycyclically presented groups. We
remark also that the methods for solving these problems in the special case
of finite p-groups, which will be discussed later in Subsections 9.4.5 and
9.4.6, are generally more efficient than those to be discussed here.
The basic approach of this method is to use induction downwards along a
characteristic series with elementary abelian factors; see Section 8.5. Thus
let
be such a series and let N=Nl be the last
nontrivial term in this series. Then N is an elementary abelian p-group of
rank d, say. By induction, we assume that we know generators and the
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order of Aut(G/N) and we aim to determine generators and the order of
Aut(G). As N is characteristic in G, there exists a mapping
where αG/N and aN denote the induced actions of a on G/N and N. We
determine Aut(G) by computing the kernel and the image of ϕ. The kernel
can be obtained from the following lemma.
LEMMA 8.59 Let N be an elementary abelian p-group.
a) Each element γ∈Z1(G/N, N) defines an automorphism αγ∈ Aut(G) by
gaγ:=gγ(gN) for g∈G.
b) The mapping
is a monomorphism
with im( )=ker(ϕ).
c) We have ker(ϕ)ХZ1(G/N, N).
PROOF a) is obvious and c) follows directly from b).
b) It is straightforward to prove that ψ is a monomorphism. It remains to
show that im(ψ)=ker(ϕ). Let α∈im(ψ). Then there exists γ∈Z1(G/N, N) such
that α=αγ. By the definition of αγ, α induces the identity on G/N and N.
Hence α∈ker(ϕ). Conversely, let α∈ker(ϕ). Then α induces the identity on
G/N and hence for each g∈G there exists ng∈N with gα=gng. As α induces
the identity on N, the function
is constant on cosets of N
and thus it is in fact a function of G/N. It is straightforward to check that δ
is a derivation of G/N and thus δ∈Z1(G/N, N). Hence α∈im(ψ) as desired.
Hence generators and the order of ker(ϕ) can be computed readily using
the methods of Section 7.6 and Lemma 8.59.
It remains to determine im(ϕ). First, we note that generators and the
order of Aut(G/N)×Aut(N) are available, since for Aut(G/N) this information
is known by induction and Aut(N)ХGL(d, p). Our aim is to determine
generators and the order for im(ϕ) from Aut(G/N)×Aut(N).
DEFINITION 8.60 Let
denote the action
homomorphism corresponding to the conjugation action of G/N on N. Then
we define the set of compatible pairs Comp(G, N) as
The following lemma notes that Comp(G, N) is a subgroup of Aut(G/N)×
Aut(N) and it also exhibits the connection between Comp(G, N) and the
image im(ϕ).
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LEMMA 8.61 im(ϕ)≤Comp(G, N)≤Aut(G/N)×Aut(N).
PROOF It is straightforward to check that Comp(G, N) is closed under
multiplication; so it is a subgroup of Aut(G/N)×Aut(N). Let (v, µ)∈im(ϕ).
Then there exists α∈Aut(G) with αG/N=v and αN=µ. For h∈G/N, we have
.
Next, we describe a method of computing Comp(G, N) from Aut(G/N)
and Aut(N). Let
be the kernel and the image
of the conjugation action of G/N on N. Then K/N=CG/N(N) and IХ(G/
N)/(K/N)ХG/K. Let S:=StabAut(G/N)(K/N) and T:=NAut(N)(I). Then Comp(G,
N)≤S×T.
Generators and the orders of S and T can be computed from Aut(G/N)
and Aut(N) using the methods of Section 4.1 and thus we can also obtain
generators and the order of S×T. The group S×T acts on the set Hom(G/
N, Aut(N)) of all homomorphisms from G/N to Aut(N) via
and the compatible pairs form the stabilizer of ψ under this action. Hence
generators and the order of Comp(G, N) can be computed from S×T using
the orbit-stabilizer method of Section 4.1.
DEFINITION 8.62 For γ∈Z2(G/N, N) and (v, µ)∈Comp(G, N) we define
It is straightforward to observe that this induces an action of Comp(G,
N) on the group of 2-cocycles Z2(G/N, N). The subgroup B2(G/N, N) is
setwise invariant under this action. Thus we obtain an induced action of
Comp(G, N) on the second cohomology group H2(G/N, N).
DEFINITION 8.63 Let γ∈Z2(G/N, N) be a cocycle defining the extension
G of N by G/N. Then we define the set of inducible pairs as Indu(G,
N)=StabComp(G, N)(γB2(G/N, N)).
By definition, the inducible pairs form a subgroup of the compatible pairs,
and they can be computed using the orbit-stabilizer algorithm of Section
4.1. This yields an algorithm to compute the image im(ϕ) by the following
theorem.
THEOREM 8.64 im(ϕ)=Indu(G, N).
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PROOF The proof of this theorem is straightforward, but technical. We
refer to [Rob81].
Theorem 8.64 yields that we can compute im(ϕ) as the stabilizer of a cocycle
coset γB2(G/N, N) under the action of the group of the compatible pairs
Comp(G, N). Hence we can compute generators and the order of im(ϕ)
using the methods of Section 4.1.
8.10 The structure of finite solvable groups
The structure of the finite solvable groups is well investigated and subgroups
such as Sylow and Hall subgroups and maximal subgroups play a major rôle
in this theory. The finite solvable groups are exactly the finite polycyclic
groups and hence every finite solvable group can be defined by a refined
consistent polycyclic presentation.
In this section we shall describe methods to compute such structure
theoretic subgroups in a group G that is defined by a refined consistent
polycyclic presentation
. For further details and more advanced
methods of a similar nature we refer to papers by Cannon, Eick, LeedhamGreen, and Wright [CELG04, Eic93, Eic97, EW02]. For background on the
underlying theory of finite solvable groups see [DH92].
8.10.1 Sylow and Hall subgroups
DEFINITION 8.65 Let p be a prime and let π be a set of primes. Write
for distinct primes p1,…, pr.
a) A Sylow p-subgroup S of G is a subgroup of p-power order such that
; that is, |G:S| is coprime to |S|.
b) A Hall π-subgroup H of G is a subgroup such that all prime divisors of
|H| are contained in π and |G:H| is coprime to |H|.
c)
A Sylow system of G is a set of Sylow subgroups {S1,…, Sr} such that Si
is a Sylow pi-subgroup and SiSj=SjSi holds for each i≠j.
d) A complement system of G is a set of subgroups {C1,…, Cr} such that
|G:Ci| is a pi-power and |G:Ci| is coprime to |Ci| for 1≤i≤r. The
group Ci is called a pi-complement.
It is well-known that every finite solvable group has a Sylow system and a
complement system. In fact, this existence can be used to characterize the
solvable groups among the finite groups. A complement system gives rise to
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Hall subgroups for every possible set of primes π and it also gives rise to a
Sylow system, as the following lemma recalls. We refer to [Hup67], VI 1.5
and VI 2.2 for a proof.
LEMMA 8.66 Let {C1,…, Cr} be a complement system of G.
a) Then
is a Hall π-subgroup in G.
b) The set {S1,…, Sr} with
forms a Sylow system for G.
It is the aim of this section to outline an algorithm for computing a complement
system in the finite polycyclic group G. For this purpose we use induction
downwards along a normal series with elementary abelian factors
. Thus let N=Nl be the last nontrivial term in
this series. Then N is an elementary abelian p-group of rank d, say.
By induction, we assume that we have determined a q-complement C/N
of G/N for some prime q. We want to compute a q-complement of G. We
have to distinguish two cases on p and q as outlined in the following lemma.
LEMMA 8.67 Let C/N be a q-complement in G/N for a p-group N.
a) If p≠q, then C is a q-complement of G.
b) If p=q, then there exists a complement to N in C and every such
complement is a q-complement of G.
PROOF a) In this case we have that |G:C| is a q-power and |C|=|C/
N||N| is coprime to q. Thus C is a q-complement in G.
b) In this case we have that |C/N| is coprime to q=p and thus |C/N|
and |N| are coprime. Hence there exists a complement to N in C by the
Schur-Zassenhaus theorem [Rot94, Theorem 7.24]. Let K be such a
complement. Then |K|=|C/N| is coprime to q and |G:K|=|G:C||N| is
a q-power. Thus K is a q-complement in G.
Lemma 8.67 yields a method to determine a q-complement by the methods of
Section 7.6. However, in this special situation, we can determine complements
with a more effective method. We give a brief outline of this improvement
in the following. It makes use of the fact that, in case b) of the lemma,
every complement of N in a subgroup of C is contained in a complement of
N in C. This follows from the other part of the Schur-Zassenhaus theorem,
which says that, when |C/N| and |N| are coprime, then all complements
of N in C are conjugate.
Let p=q and let Y:=[c1,…, cr, n1,…, nd] be a polycyclic sequence for C such
that
and such that the relative orders R(Y)=(s1,…, sr, p,…,
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p) are all primes. Then every complement T of N in C has a polycyclic
sequence of the form [c1l1,…, crlr] for certain elements li∈N. Our aim is to
determine such elements l 1,…, l r . The following lemma yields a
characterization of these elements.
LEMMA 8.68 Let Z:=[c1l1,…, crlr] for l1,…, lr∈N. Then Z is an induced
polycyclic sequence for a complement to N in C with respect to Y if and only
if (cjlj)sj and the commutators [cjlj, cklk] can be written as normal words in Z
for 1≤j
PROOF This follows directly from Lemma 8.34.
By induction, we assume that the elements lj+1,…, lr have already been
determined and we want to compute lj. Let bk:=cklk for j+1≤k≤r, and let
Y′:=[c1,…, cj, bj+1,…, br, n1,…, nd]. Then Y′ is an induced polycyclic sequence
for C. Therefore, by Lemma 8.34, we find that the power
and the
commutators [bk, cj] for j+1≤k≤r can be written as normal words in Y′ of
the form
The following lemma yields a method to determine lj from this setup.
LEMMA 8.69 Let C be a group and
an abelian normal subgroup.
a) Let c,b∈C with [b, c]=wv for v∈N. Then, for
only if lα=v with a=bw-1.
b) Let c∈C with cs=wv for v∈N. Then, for
lb=v with
.
if and
if and only if
PROOF The proof is an elementary computation. We refer to [CELG04] for
an outline.
8.10.2 Maximal subgroups
In this section we shall give an outline of an effective algorithm to compute
the conjugacy classes of maximal subgroups of G.
DEFINITION 8.70
a) N/L is called a normal factor of G if L,
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with L≤N.