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7 Sylow subgroups, p-cores, and the solvable radical

7 Sylow subgroups, p-cores, and the solvable radical

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is defined to be the largest normal solvable subgroup of G. Although the pcore was at one time computed as the intersection of the Sylow p-subgroups

of G, this turned out not to be a good approach, since it involves backtracks

in both the Sylow subgroup and intersection phases.

It is shown, for example, by Luks in [Luk93], that both Op(G) and O∞(G)

are computable in polynomial time, and nearly-linear-time algorithms are

described in Section 6.3.1 of [Ser03]. Fast practical algorithms are also available

for computing these subgroups. The approach that we shall describe here

for Op(G) is due to Unger [Ung02].

We have already (in Subsection 3.1.1) remarked on the unfortunate fact

that we cannot in general find a faithful permutation representation of a

quotient group G/N of a permutation group G of degree comparable to that of

G. However, it turns out that we can find such a representation for G/

Op(G) and for G/O∞(G). This result was first proved by Easdown and Praeger

in [EP88] and is also proved by Holt in [Hol97], where it is shown that the

homomorphisms involved can be computed explicitly. We shall include a

proof of it here. It enables us to compute O∞(G) by first finding some

nontrivial Op(G) and then solving the problem recursively in G/Op(G).

4.7.1 Reductions involving intransitivity and imprimitivity

We can reduce the difficulty of many computational problems in finite

permutation groups by a divide-and-conquer technique using actions on orbits,

in the case of intransitive groups, and actions on blocks, in the case of

imprimitive groups.

Some of these reductions are based on the idea of solving the problem, in

turn, in two proper quotients G/M and G/N of G, where M N=1. More

precisely, we have the following results for Sylow subgroups and Op(G).

LEMMA 4.13 Let G be a finite group having normal subgroups M and N

with M N=1 and let µ:G→G/M and v:G→G/N be the natural epimorphisms.

Let p be a prime.

(i) Let Q/M∈Sylp(G/M) and Q:=µ-1(Q/M), and let

and

Then P∈Sylp(G).

(ii) Let Q/M=Op(G/M) and Q:=µ-1(Q/M), and let

Then P=Op(G).



and



PROOF (i) Clearly Q contains a Sylow p-subgroup of G and, since M N=1,

Q N is a p-group. Similarly, R contains a Sylow p-subgroup of Q and hence

of G. Now R=PN, where

is a p-group. Furthermore

P N=Q N, so P N and hence P is a p-group, and the result follows.



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(ii) As in (i), P N=Q N is a p-group, so P=Op(Q). But O p(G)≤Q, so

Op(G)=Op(Q)=P.

There are two specific situations in which we will apply this lemma. The first

is when GΩ is intransitive, in which case we can write Ω=Ω1∪Ω2 where G fixes

the nonempty and nonintersecting sets Ω1 and Ω2. As we saw in Subsection

4.5.1, we can compute easily and efficiently with the induced action

homomorphisms

and

. Let M:=ker(µ) and N:=ker(v).

Then, since G is a permutation group on Ω=Ω1∪Ω2, we have M∩N=1.

We can then recursively call our algorithm for computing Op(G) or a

Sylow p-subgroup of G on

and

, which both have

smaller permutation degree than G. Then we can use the recipe in Lemma

4.13 to construct Op(G) or a Sylow p-subgroup of G.

The second situation is when GΩ is imprimitive and has two distinct minimal

block systems Σ1 and Σ2, say. Of course, to apply this reduction, we first need

to apply the algorithms described in Section 4.3 to find all such minimal

block systems. Here, we let

and

be the induced

action homomorphisms. We saw in Subsection 4.5.2 that we can compute

effectively with these homomorphisms. Again, we must have M∩N=1,

because otherwise the orbits of M∩N would form a block system with blocks

strictly contained in the blocks of Σ1 and Σ2, thereby contradicting their

minimality. So again we can solve our problems by using recursive calls to

smaller examples.

In the case of a unique minimal block system Σ, we can still use recursion

on GΣ, which enables us to assume that GΣ is a p-group, but we cannot make

any immediate further reductions. (A p-group is, by definition, a group of

order pn for some n≥0.)

In the following two subsections, we shall consider in more detail the

cases of computing Sylow p-subgroups and Op(G). It will turn out that, for

Op(G), the unique minimal block system obstacle does not arise.



4.7.2 Computing Sylow subgroups

As explained in the previous subsection, when computing a Sylow p-subgroup

of G, we can assume, by recursion and part (i) of Lemma 4.13, that G is

transitive, and is either primitive, or is imprimitive with a unique minimal

block system Σ such that GΣ is a p-group. We can also assume that |Ω| is

divisible by p, since otherwise we just compute a Sylow p-subgroup of the

smaller group Gα for some α∈Ω.

There are other possible reductions we could make. One approach, which

is advocated in [CCH97b], is to solve the Sylow subgroup finding algorithm

at the same time as solving the problem of finding a conjugating element for

two Sylow subgroups P and Q of G. This allows us to make further reductions

in the imprimitive case, by using mutual recursion between the two



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problems. Another possibility, which we shall mention again in a later

chapter, is to compute O∞(G) and, if it is nontrivial, recursively compute

R/O∞(G)∈Sylp(G/O∞(G)), and then find a Sylow p-subgroup of the solvable

group R, possibly using a PC-representation.

But however many recursions we make, if our group is primitive and

non-abelian simple, then we can make no more of these types of reductions.

In the polynomial-time algorithm of Kantor, we would first need to identify

the isomorphism type of the simple group. This is not difficult and, by

Proposition 10.8, it can nearly always be done just from a knowledge of |G|.

In the few ambiguous cases, there are simple tests to distinguish between

the two possibilities. We then need to set up an isomorphism between G and

the natural representation of the isomorphic simple group. For example, if

, then we would define an epimorphism from the natural

representation of SL(n, q) as a group of d×d matrices over to G. We then

simply ‘write down’ a Sylow subgroup in the natural representation, and

compute a Sylow subgroup of G by using the isomorphism.

There is one situation in which we do this without any further work at all,

namely when G is the alternating or symmetric group on Ω, when it is

straightforward to simply write down a Sylow p-subgroup, so we should

certainly do that.

Otherwise, in the absence of the relatively large amount of machinery

required to implement the Kantor approach, which we have only hinted at

here, we are forced to fall back on backtrack algorithms. The most successful

approach to this has been the method of Butler and Cannon, which we shall

describe here. The backtrack part of the method occurs in the calculation of

centralizers of group elements. Other possible approaches use normalizer

calculations, but they are typically significantly slower than centralizer

calculations.

The basic aim of the Butler-Cannon method is to find an element g∈G of

order p such that C:=CG(g) contains a Sylow p-subgroup of G. Once we have

done that, and calculated C, then we apply recursion to C and thereby assume

that C=G. Now g has order p, so the orbits of g have size 1 or p. Since we are

assuming that GΩ is transitive, and, by Proposition 4.9, G permutes the

orbits of g, all of the orbits of g must have order p, and they form a minimal

block system Σ of GΩ. We are also assuming that GΣ is a p-group. We claim

that G is p-group, and so in fact we are done.

To prove this, we must show that the kernel K of the action of G on Σ is a

p-group. Let Ωi (1≤i≤r) be the orbits of g and, for 1≤i≤ r, let gi be the

permutation of Ω that acts like g on Ωi, and fixes all points of Ωj for j≠i. Then

the elements gi generate an elementary abelian p-group Q of order pr, which

contains g. We claim that K≤Q. To see this, let h∈K. Since K≤C,

centralizes

for each i, and so, by Corollary 4.11,

for

some mi with 0≤mi
, which proves the claim.

There remains the problem of finding a suitable element g. We start by



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SYLOW(G, p)



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Input: A permutation group G and a prime p

Output: Generators of P∈Sylp(G)

if G is a p-group then return G;

if GΩ is intransitive then recurse on orbit actions;

if p does not divide |Ω| then recurse on Gα;

if GΩ is imprimitive with two minimal block systems

then recurse on block actions;

if GΩ is imprimitive with unique minimal block system Σ and

GΣ is not a p-group

then recurse on GΣ;

if G=Sym(Ω) or Alt(Ω) then return a Sylow p-subgroup;

Find an element g of G of order p;

if p2 does not divide |G| then return ;

Compute C := CG(g);

while C does not contain a Sylow p-subgroup of G

do Find S∈Sylp(C) and compute Z :=Z(S);

for elements h∈Z with |h|=p do compute Ch:=CG(h)

until we find an h such that p divides |Ch/S|;

Replace C by Ch;

Recurse on C;



looking for any element g of order p. To do this, we choose random elements

h of G until we find one whose order pm is multiple of p, and take g:=hm. In

many groups this works very quickly. There are a few examples in which it

does not; for example, there are groups of order p(p-1) in which only p-1

elements have order a power of p, which is a small proportion when p is

large. In this, and other similar examples, a policy of sometimes choosing h

to be a commutator of random elements already considered will often find the

required element quickly.

Having found g of order p we must then compute C:=CG(g). If C contains

a Sylow p-subgroup of G then we are done. Otherwise, we still compute

S∈Sylp(C). Now, by Sylow’s theorem, S is contained in some P∈Sylp(G),

and P has nontrivial centre, so there exists h∈Z(P) with h of order p. Since

h∈C, we must have h∈S, because otherwise S would be properly contained

in P∩C, contradicting S∈Sylp(C). So we know that there exists an element

g of order p in Z(S) having the required property that CG(g) contains a

Sylow p-subgroup of G.

So we compute Z(S), which is not difficult, and then search through the

elements g of order p (see Exercise 3 below) in Z(S) looking for the required

g. This necessitates computing CG(g) for each such g, and is potentially the

slowest part of the algorithm because, in a bad case, there could be many

such elements g, with only a small proportion of them having the required

centralizer. In practice, if we find any g for which the Sylow p-subgroup of



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CG(g) is larger than S, then we replace S with this new Sylow p-subgroup.

SYLOW is a summary of the complete Sylow p-subgroup algorithm.



4.7.3 A result on quotient groups of permutation groups

In this subsection, we shall prove the following result, which was probably

first proved by Easdown and Praeger in [EP88]. The specific classes of

groups to which we shall apply the result are the classes of all abelian groups,

and of all elementary abelian p-groups for some prime p.

THEOREM 4.14 Let G be a finite permutation group on Ω of degree n. Let

A be a class of finite abelian groups closed under subgroup, quotient, and

direct product, and let N be a normal A-subgroup of G. Then there exists a

permutation representation ϕ of G of degree at most n, such that the kernel

K of ϕ is an A-group that contains N.

PROOF We may clearly assume that N≠1. Let S :=Sym(Ω), let Σ be the set

of orbits of N on Ω, and let Γ be a subset of Ω containing one representative

from each orbit ∆∈Σ. Let L be the subgroup of S consisting of all g∈S such

that g fixes each ∆∈Σ and, for all ∆∈Σ, there exists h∈N with g∆=h∆. Then

L is isomorphic to the direct product of the induced action groups N∆ for

∆∈Σ, and so, by the assumptions on the class A, we ave L∈A. It is easy to

check that L is normalized by G. Let E:=GL, and let K:=G∩L. Then clearly

N≤K, K∈A (since A is subgroup closed), and

.

Let D:=EΓ be the setwise stabilizer of Γ in E. We claim that E=DL. Let

e∈E. To establish the claim, we shall define an element f∈L such that ef ∈D.

To do this, we need to define f∆ for each ∆∈Σ. Since G permutes the set Σ, we

have

and there is a unique point

. Then γe∈∆, and so

eh

there exists h∈N with γ equal to the unique point in Γ∩∆. We define f∆ to

be equal to h∆. Then, by definition of L, we have f∈L, and by choice of h, we

have γef∈Γ for all γ∈Γ. Thus ef∈D, and E=DL as required. Since L is abelian,

by Corollary 4.11, any element in L that fixes a point of one of its orbits fixes

the whole orbit pointwise. But L∩D must fix every point in Γ, and hence

L∩D=1, and D is a complement of L in E.

Now we have

, and since D is a permutation

group of degree n, we can define the required permutation representation ϕ

of G by ϕ(g)=gf∈D, where f is as defined above for e=g. This completes the

proof.

For computational applications, it is important to observe that the proof

allows us easily to compute ϕ(g)=gf for g∈G, and hence we can compute

effectively with the homomorphism ϕ.



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COROLLARY 4.15 Let G be a finite permutation group on Ω of degree n.

Then there exist faithful permutation representations of degree at most n of

G/Op(G), for any prime p, and of G/O∞(G).

PROOF The proof is by induction on |G|. The results are all clear if

O∞(G)=1, so assume not, and let N be a minimal normal subgroup of G

contained in Op(G) or in O∞(G), as appropriate. Since O∞(G) is solvable, its

composition factors are all cyclic groups of prime order, and it follows

from Corollary 2.32 that N is an elementary abelian p-group. So, by

Theorem 4.14, there is an abelian group K containing N such that G/K

has a permutation representation of degree at most n, and the result

follows by induction applied to G/K.

Provided that we can compute Op(G) for primes p, which we shall be

explaining how to do in the next subsection, then the representations in

Corollary 4.15 can again be computed explicitly. For the representation of

G/Op(G), we could, for example, first compute Op(G), then N := Z(Op(G)),

then use Theorem 4.14 to find a faithful permutation representation ϕ1 of

G/K1 where N≤K1≤Op(G). Then, if K1≠Op(G), we repeat the process on

im(ϕ 1) to get a faithful permutation representation of G/K 2 where

K 1
representation ϕ of G/Op(G). (Notice that it is neither necessary nor

desirable to choose N to be elementary abelian like we did in the proof of

the corollary.)

To get the faithful representation of G/O∞(G), we first find a prime p

with Op(G)≠1, then find a faithful permutation representation ϕ1 of G/

Op(G), as described above. Then, if Op(G)≠O∞(G), we repeat the process

on im(ϕ1), and carry on until like this until we have found the required

representation.



4.7.4 Computing the p-core

The method that we shall describe here for Op(G) is due to Unger [Ung02].

As was the case for Sylow p-subgroups, we can reduce, by recursion and part

(ii) of Lemma 4.13, to the situation where G is transitive, and is either

primitive, or is imprimitive with a unique minimal block system Σ such

that GΣ is a p-group.

Let us first deal with the unique minimal block system case. Let K be the

kernel of the action of G on Σ. If K=1, then G is a p-group and G=Op(G), so

assume not. We recursively compute Q :=Op(K). Suppose that Q=1. Then

we claim that Op(G)=1. If not, then let N be a minimal normal subgroup of

G contained in Op(G). Since the composition factors of any p-group are

cyclic of order p, it follows from Corollary 2.32 that N is an elementary

abelian p-group. Since Q=1, we have K∩N=1, and hence [K,N]≤K∩N=1, so



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K≤ CG(N). Let ∆ be an orbit of NΩ. Then, by Proposition 2.29, either ∆=Ω,

or ∆ is a block of imprimitivity of GΩ. Since Σ is the unique minimal block

system of GΩ, it is true in either case that ∆ is a union of blocks of Σ. So K

fixes the set ∆, and then Corollary 4.11 implies that K∆≤N∆ and hence K∆

is a p-group. But this is true for all orbits ∆ of K, so K is a p-group,

contradicting Q=1. Hence Op(G)=1 as claimed, and so we may assume that

Q=Op(K)≠1.

Now we compute N:=Z(Q) and use Theorem 4.14 to construct a

epimorphism ϕ:G→H for which M:=ker(ϕ) is a p-group containing N. Then

we can apply recursion to compute Op(H) and hence Op(G):=ϕ—1(Op(H)).

It remains to deal with the case in which GΩ is primitive. In that case, if

Op(G)≠1, then let N be a minimal normal subgroup of G contained in Op(G).

As above, N is elementary abelian, and NΩ is transitive by Proposition 2.29

and regular by Corollary 4.11. So it is an elementary abelian regular normal

subgroup, which is often abbreviated to EARNS, of G.

If N
have N≤Z(Op(G)), so 1
as a normal Subgroup of G. Hence N=Op(G).

So the problem reduces to deciding whether a primitive group has an

EARNS. The method that we shall describe for this comes originally from

P.M. Neumann [Neu87]. If the degree n of G is not equal to a prime power pd,

then clearly G has no EARNS, so we may assume that n=pd.

First we need to recall some basic properties of Frobenius groups.

DEFINITION 4.16 A Frobenius group is a transitive permutation group on

a set Ω such that Gα≠1, but Gαß=1 for all distinct α, ß∈Ω.

Let GΩ be a finite Frobenius group of degree n. By a simple counting argument,

we find that there are exactly n-1 elements of G that fix no point of Ω. The

fundamental result of Frobenius, dating from 1901, is that these n-1

elements, together with the identity, form a normal subgroup K of G, known

as the Frobenius kernel of G. The proof uses character theory, and no proof

is currently known that does not use character theory. The stabilizer Gα is

known as the Frobenius complement. Various properties of the Frobenius

complement can be shown without too much difficulty. For example, the

centre is nontrivial, the Sylow p-subgroups are cyclic for p odd, and cyclic or

generalized quaternion for p even, and A5 is the only possible nonabelian

composition factor of a Frobenius complement. A much deeper result, due to

J.G. Thompson (1959), is that the Frobenius kernel is nilpotent. For proofs of

all of these properties see, for example, [Pas68].

Returning to our EARNS algorithm, we first check whether Gαß=1 for all

distinct α, ß∈Ω. If so, then either Gα=1, in which case GΩ primitive implies

that |G|=p, so G itself is the EARNS, or G is a Frobenius group. In that

case, we compute Z=Z(Gα) for some α, which we know is nontrivial, choose

z∈Z\{1}, choose h∈G\Gα, and let g:=[h, z]. Then g≠1 is in the Frobenius



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kernel of G (exercise) and, since GΩ is primitive, we can compute the full

kernel, which is the required EARNS, as the normal closure

.

So we assume that Gαß≠1 for some distinct α, ß∈Ω. We fix α and then

choose ß from the orbit of Gα on Ω\{α} of the smallest size. This gives the

largest possible |Gαß|, so clearly Gαß≠1 with this choice. Let ∆ be the set of

fixed points of Gαß; that is,

.

Suppose that G does have an EARNS N. Then, for γ∈∆, there is a (unique)

g∈N with αg=γ. For h∈Gαß, we have g—1hg∈Gγ, so [h, g]∈N∩Gγ=1, and

hence g∈C := CG(Gαß). In particular, N∩C acts regularly on ∆, so if |∆| is

not a power of p, then G has no EARNS, and we are done.

Otherwise, we compute C. This is typically an easy centralizer computation,

but it can be done in polynomial (indeed, in almost linear) time; see Section

6.2.3 of [Ser03] for details. If C does not act transitively on ∆, then again G

has no EARNS, so assume that it does.

For any γ∈∆\{α}, we have Gαß≤Gαγ by definition of ∆, but the the choice

of β forces Gαβ=Gαγ, so Gαγ fixes all points of ∆, and C∆ must be regular or a

Frobenius group. In the latter case, (N∩C)∆ must be the Frobenius kernel.

So we compute this kernel K∆ as described above for the case when GΩ was a

Frobenius group, and then let K be the inverse image of K∆ in C. (There is an

extra complication here, because C∆ might be an imprimitive Frobenius group,

but we can choose further generators of the form g:=[h, z] for z∈Z(Cα) and

random h∈C\Cα until we have generated the whole of K.) If C∆ is regular,

then we just let K=C.

In either case, we now have a subgroup K≤C with K∆ regular which, if

the EARNS N exists at all, satisfies N∩C≤K and (N∩C)∆=K∆. Next we

compute the subgroup P:={x∈Cαβ|xp=1}; see Exercise 3 below. Choose any

x∈C with |x|=p and x∉Cαβ. (Any 1≠x∈N∩C has this property, so if there is

no such x, then G has no EARNS.)

Then x∆=g∆ for some g∈N∩C, so x—1g∈P. Hence we can find g as xh for

some h∈P. For each such xh, we test whether g:=xh∈N by computing

,

and checking whether it is an EARNS. If we try all h∈P and find no EARNS,

then G has no EARNS. It can be shown (see exercises below) that if N exists,

then |P≤ pd—1 (recall that n=|Ω|=pd), so there are not too many elements to

try! As a refinement, if, on calculating P, we find that |P|>pd—1, then there

can be no EARNS, and we can stop immediately.

P-CORE is a summary of the complete p-core algorithm.



4.7.5 Computing the solvable radical

It is now straightforward to compute the solvable radical O∞(G) of G. We

consider the primes p dividing |G| and compute Op(G) as described above.

If Op(G)=1 for all such p, then O∞(G)=1.

If we find a prime p with Op(G)≠1 then, by Corollary 4.15, we can compute

an explicit epimorphism ϕ:G→H:=G/O p(G). Now we can recursively

compute O∞(H) and hence compute O∞(G):=ϕ-1(O∞(H)).



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P-CORE(G,p)



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Input: A permutation group GΩ and a prime p

Output: Generators of Op(G)

if G is a p-group then return G;

if GΩ is intransitive then recurse on orbit actions;

if GΩ is imprimitive with two minimal block systems

then recurse on block actions;

if GΩ is imprimitive with unique minimal block system Σ

then Compute the kernel K of the action of G on Σ;

if K=1 then recurse on GΣ;

Compute Q :=Op(K);

if Q=1 then return {};

Compute N :=Z(Q);

Find a permutation representation ϕ of G such that

M:= ker(ϕ) is a p-group with N≤M;

Recursively compute L:=Op(im(ϕ));

return generators of ϕ—1 (L);

(* Now GΩ is primitive—we use Neumann’s EARNS routine *)

if n :=|Ω| is not a power pd of p then return {};

Choose α∈Ω;

if GΩ is a Frobenius group

then Choose 1≠z∈Z(Gα) and h∈G\Gα;

return generators of

;

Choose ß in an orbit of Gα on Ω\{α} of minimal length;

Compute the fixed point set ∆ of Gαβ;

if ∆ is not a power of p then return {};

Compute C := CG(Gαβ);

If C∆ is not transitive then return {};

Compute subgroup R≤C, where R∆ is the regular normal subgroup of

the regular or Frobenius group C∆;

Compute P := {x∈Cαβ|xp=1};

if |P|>pd—1 then return {};

Find x∈R\Caß with |x|=p;

for h∈P

do Compute

;

if N is an EARNS of G then return generators of N;

return {};



4.7.6 Nonabelian regular normal subgroups

In this final section, we shall take note of the fact that Neumann’s EARNS

algorithm can be used almost unchanged to test for the existence of, and find

nonabelian regular normal subgroups in a primitive permutation group GΩ.

We shall be using this in Subsection 10.1.4 as part of an algorithm to find

the socle of a general primitive permutation group.



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By Thompson’s result, a Frobenius kernel is nilpotent, and so the Frobenius

kernel of a primitive group is elementary abelian, and hence a primitive

Frobenius group cannot have a nonabelian regular normal subgroup.

As in the EARNS routine, we fix α∈Ω and choose β from the orbit of Gα

on Ω\{α} of the smallest size; then Gαβ≠1. Let ∆ be the set of fixed points of

Gαβ, and compute C:=CG(Gαβ).

As in the EARNS case, for any regular normal subgroup N of G, N∩C acts

regularly on ∆, so if C∆ is not transitive, then there is no such N, and we are

done. By choice of ß, C∆ must either be a Frobenius group or act regularly. In

the first case we find K≤C such that K∆ is the Frobenius kernel, and in the

second case we let K=C.

Choose g∈K of prime order p, with x∆≠1 (if there is no such g, then there

is no regular normal subgroup), and compute the subgroup P consisting of

elements of order 1 or p in Cαβ. Then, as in the EARNS case, a regular

normal subgroup of G is the normal closure of xh for some h∈P, and so we

just try all possible h.

We shall see later, in Theorem 10.1.3, that a primitive group can have one

or two nonabelian regular normal subgroups. If we find one such subgroup

N, then we could either continue the search through h∈P to look for a possible

second subgroup, or use the fact that this second subgroup, if it exists, must

be the centralizer in Sym(Ω) of N.

It can be shown, using the O’Nan-Scott theorem (Theorem 10.1.3), that

|P|<|Ω|, and so this algorithm can be made to run in polynomial time.

Seress points out in Remark 6.2.14 of [Ser03] that it is not almost linear, and

he presents an alternative almost linear Monte Carlo algorithm due to Luks

to find nonabelian regular normal subgroups.



Exercises

Assume that G is finite in these exercises.

1. Prove that Op(G) is the intersection of the Sylow p-subgroups of G.

2. Prove that O∞(G)/≠1 if and only if Op(G)≠1 for some prime p.

3. Let 〈G=〈X〉 be a finite abelian p-group. Devise an algorithm for computing

the subgroup P of G consisting of all elements of order 1 or p.

(Hint: Let

each i, if



with oi≥oi+1 for 1≤i
, then include



P. Otherwise, replace xi by xig for suitable



as a generator of

to reduce



|xi|.)

4. Work out how to write down generators of a Sylow p-subgroup of Sym(n)

using the following steps.



© 2005 by Chapman & Hall/CRC Press



Computation in Finite Permutation Groups



(i)



143



First write n in base p. That is,

with 0≤ai
group orders, show that a Sylow p-subgroup of Sym(n) is contained

in the direct product of groups Xi (1≤i≤r), where each Xi is itself

the direct product of ai copies of Sym(pi). This effectively reduces

the problem to the case when n is a power pi of p.



(ii) When n=pi show that a Sylow p-subgroup of Sym(n) is generated

by a Sylow p-subgroup of Sym(pi—1) acting on the set {1..pi—1}

together with the permutation g∈Sym(n) with jg=j+pi—1 (mod pi)

for 1≤ j≤pi.

(For Pi∈Sylp(Sym(pi)), we have

and, for i>1, Pi is the wreath

product

; see, for example, [Hal59, Sec. 5.9].)

5. Modify the method of Exercise 4 to write down P∈Syl2(Alt(n)).

6. Show that there are exactly n-1 fixed-point-free elements in a Frobenius

group of degree n.

7. In the description of the Op(G) algorithm for Frobenius groups, prove

the claim that the element g defined as [h, z] is a nontrivial element in

the Frobenius kernel.

8. Let H be any finite group and p a prime with Op(H)=1. Let P be an

abelian p-subgroup of H, and C := CH(P). Prove that |P|≤|H:C|.

(Hint: Use induction on |H:C|. Let Q be a conjugate of P in H that

does not centralize P. Let P* :=P∩CH(Q), C* :=CH(P*), and deduce by

induction that |P*|≤|H:C*|. Assume WLOG that |P*|≥|Q∩CH(P)|,

and deduce |C*:C|≥|P:P*|.)

9. We saw earlier that, if a primitive group GΩ has an EARNS N, then

N=Op(G) and hence Op(Gα)=1. Taking P to be the elements of order

dividing p in Cαβ with C:=CG(Gαβ) as in P-CORE, apply the result in the

previous exercise to H=Gα to prove that |P|≤pd-1.



4.8 Applications

In this section we briefly review three applications of the methods discussed

in the chapter. The first, to card shuffling, is primarily recreational, but has

some relevance to computer networks. The second is to the construction of

various types of combinatorial objects, including error-correcting codes, which

are important outside of mathematics. The third is another application to

computer networks.



© 2005 by Chapman & Hall/CRC Press



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