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8 Intersections, centralizers, and normalizers

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.



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