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