Jordan-Holder theorem for imprimitivity systems and maximal decompositions of rational functions

0.29 MB·English

by  M. Muzychuk

JORDAN-HO¨LDER THEOREM FOR IMPRIMITIVITY SYSTEMS
AND MAXIMAL DECOMPOSITIONS OF RATIONAL
FUNCTIONS
9
0
0 M.MUZYCHUK,F.PAKOVICH
2

n Abstract. Inthispaperweproveseveralresultsaboutthelatticeofimprim-
a itivity systems of a permutation group containing a cyclic subgroup with at
J mosttwoorbits. AsanapplicationwegeneralizethefirstRitttheoremabout

7 functionaldecompositionsofpolynomials,andsomeotherrelatedresults. Be-
sides,wediscussexamplesofrationalfunctions,relatedtofinitesubgroupsof
] Aut(CP1),forwhichthefirstRitttheoremfailstobetrue.
V
C
.
h 1. Introduction
t
a
m Let F be a rationalfunction with complex coefficients. The function F is called
indecomposable ifthe equalityF =F ◦F ,whereF ◦F denotesthesuperposition
[ 1 2 1 2
F (F (z)) of rational functions F , F , implies that at least one of the functions
1 2 1 2
2 F ,F is of degree 1. A rational function which is not indecomposable is called
1 2
v decomposable. Any representation F of a rational function F in the form
9
6 (1) F =F ◦F ◦···◦F ,
1 2 r
8
where F ,F ,...,F are rational functions, is called a decomposition of F. If all
3 1 2 r
. F1,F2,...,Fr are indecomposable of degree greater than one, then the decomposi-
2 tion F is called maximal. Two decompositions of a rational function F
1
7 (2) F =U ◦U ◦···◦U and F =V ◦V ◦···◦V ,
1 2 k 1 2 m
0
: maximal or not, are called equivalent if they have the same length (that is k =m)
v
and there exist rational functions of degree one µ , 1≤i≤k−1, such that
i i
X U =V ◦µ , U =µ−1 ◦U ◦µ , 1<i<k, and V =µ−1 ◦V .
1 1 1 i i−1 i i k k−1 k
r
a Inthepaper[30]Rittdescribedthestructureofpossiblemaximaldecompositions
of polynomials. This description can be summarized in the form of two theorems
usually called the first and the second Ritt theorems (see [30], [33]). The first Ritt
theorem states that for any two maximal decompositions D,E of a polynomial F
there exists a chain of maximal decompositions F , 1 ≤ i ≤ s, of F such that
i
F = D, F ∼ E, and F is obtained from F , 1 ≤ i ≤ s−1, by replacing two
1 s i+1 i
successive functions in F by other functions with the same composition. This
i
implies in particular that any two maximal decompositions of a polynomial have
the same length. Belowwe willcalltwomaximaldecompositionsD,E ofa rational
function F such that there exists a chain as above weakly equivalent. This defines
an equivalence relation on the set of maximal decompositions of F.
1991 Mathematics Subject Classification. Primary20E15;Secondary 30D05.
Keywordsandphrases. Jordan-Ho¨ldertheorem,subgrouplattices,Ritt’stheorems,decompo-
sitionsofrationalfunctions.
1
2 M.MUZYCHUK,F.PAKOVICH
The first Ritt theorem reduces the description of maximal decompositions of
polynomialstothedescriptionofindecomposablepolynomialsolutionsoftheequa-
tion
(3) A◦C =B◦D
such that the decompositions A◦C and B◦D are non-equivalent, and the second
Ritt theorem states if A,B,C,D is such a solution then there exist polynomials
Aˆ,Bˆ,Cˆ,Dˆ, µ ,µ , where degµ =1,degµ =1, such that
1 2 1 2
A=µ ◦Aˆ, B =µ ◦Bˆ, C =Cˆ◦µ , D =Dˆ ◦µ , Aˆ◦Cˆ =Bˆ◦Dˆ,
1 1 2 2
and up to a possible replacement of Aˆ by Bˆ and Cˆ by Dˆ either
Aˆ◦Cˆ ∼zn◦zrR(zn), Bˆ◦Dˆ ∼zrRn(z)◦zn,
where R(z) is a polynomial, r≥0, n≥1, and gcd(n,r)=1, or
Aˆ◦Cˆ ∼T ◦T , Bˆ◦Dˆ ∼T ◦T ,
n m m n
where T ,T are the corresponding Chebyshev polynomials, n,m ≥ 1, and
n m
gcd(n,m) = 1. Furthermore, the second Ritt theorem remains true for arbitrary
polynomial solutions of (3) if to replace equalities degµ = 1,degµ = 1 by the
1 2
equalities
degµ =gcd(degA,degB), degµ =gcd(degC,degD)
1 2
(see [9], [34]).
Notice that the classification of polynomial solutions of (3) appears in a variety
of different contexts some of which are quite unexpected. For example, this classi-
ficationis closely relatedto the problemof descriptionofDiophantine equations of
the form A(x) = B(y), A,B ∈Z[z], having an infinite number of integer solutions
(see [10], [5]), and to the problem of description of polynomials C,D satisfying the
equality C−1{S} = D−1{T} for some compact sets S,T ⊂ C, recently solved in
[24]. Notice also that the problem of description of solutions of (3) such that C
andD arepolynomialswhileA,B areallowedtobearbitraryrational(orevenjust
continuous) functions on the sphere can be reduced to the description of polyno-
mialsolutions(see [25]). A moredetailed accountofdifferent resultsrelatedto the
second Ritt theorem can be found in the recent papers [26], [28].
The classification of polynomial solutions of (3) essentially reduces to the de-
scription of polynomials A,B such that the algebraic curve
(4) A(x)−B(y)=0
has an irreducible factor of genus zero with one point at infinity. On the other
hand, the proof of the first Ritt theorem can be given in purely algebraic terms
whichdonotinvolvethegenusconditioninanyform. Indeed,ifG(F)≤Sym(Ω)is
the monodromygroupofa rationalfunctionF thenequivalenceclassesofmaximal
decompositions of F are in a one-to-one correspondence with maximal chains of
subgroups
(5) G (F)=T <T < ... <T =G(F),
ω 0 1 r
where G (F) is the stabilizer of an element ω ∈ Ω in the group G(F). Therefore,
ω
any two maximal decompositions of F are weakly equivalent if and only if for
any two maximal chains of subgroups as above R , R there exists a collection of
1 2
maximal chains of subgroups T , 1 ≤ i ≤ s, such that T = R , T = R , and
i 1 1 s 2
JORDAN-HO¨LDER THEOREM FOR IMPRIMITIVITY SYSTEMS 3
T is obtained from T , 1 ≤ i ≤ s−1, by a replacement of exactly one group.
i+1 i
It was shown in the paper [21] (Theorem R.3) that the last condition is satisfied
for any permutation group G containing an abelian transitive subgroup. Since the
monodromygroupofapolynomialalwayscontainsacyclicsubgroupwithoneorbit
(its generator corresponds to the loop around infinity), this implies in particular
the truth of the first Ritt theorem for polynomials.
It was also proved in the paper [21] (Claim 1) that if A,B,C,D are indecom-
posable polynomials satisfying (3) such that the decompositions A◦C and B◦D
are non-equivalent then the groups G(A) and G(D) as well as the groups G(C)
and G(B) are permutation equivalent. Since any two maximal decompositions of
a polynomial P are weakly equivalent, this implies by induction that for any two
maximal decompositions (2) of P there exists a permutation σ ∈S such that the
k
monodromy groups of U and V , 1 ≤ i ≤ k, are permutation equivalent ([22]).
i σ(i)
The algebraiccounterpartof this fact is the following statement: if G≤Sym(Ω) is
a permutation group containing a cyclic subgroup with one orbit then for any two
maximal chains
G =A <...<A =G and G =B <...<B =G
ω 0 k ω 0 m
the equality k = m holds and there exists a permutation σ ∈ S such that the
k
permutation group induced by the action of Ai on cosets of Ai−1 is permutation
equivalent to the permutation group induced by the action of B on cosets of
σ(i)
Bσ(i)−1, 1 ≤ i ≤ k. If a permutation group G satisfies this condition, we will say
that G satisfies the Jordan-Ho¨lder theorem for imprimitivity systems.
Inthis paper,we extendthe aboveresultsaboutthe permutationgroupsGcon-
taining acyclic groupwith one orbitto the permutationgroupscontaininga cyclic
subgroup H with at most two orbits and apply these results to rational functions
(or more generally to meromorphic functions on compact Riemann surfaces) the
monodromy group of which contains H.
First,weprovethatforapermutationgroupGcontainingH thelatticeL(G ,G),
ω
consisting of subgroups of G containing G , is lower semi-modular and even a
ω
stronger condition of the modularity of L(G ,G) holds whenever L(G ,G) does
ω ω
not contain a sublattice isomorphic to the subgroup lattice of a dihedral group. It
follows easily from the lower semimodularity of L(G ,G) that one can pass from
ω
any chain of subgroups (5) to any other such a chain by a sequence of replace-
ments as above and therefore the first Ritt theorem extends to rational functions
the monodromy group of which contains H. Notice that this implies in particular
that the first Ritt theorem holds for rational functions with at most two poles.
Althoughfor suchfunctions the resultwasknow previously(see [27], [28], [36]) the
algebraicproof turns out to be more simple and illuminating. Notice also that our
description of the lattice L(G ,G) for groups G containing H has an interesting
ω
connection with the problem of description of algebraic curves having a factor of
genus zero with at most two points at infinity, studied in [10], [5].
Further, we prove that if a permutation group G contains a cyclic subgroup
with two orbits of different length then the lattice L(G ,G) is alwaysmodular and
ω
G satisfies the Jordan-Ho¨lder theorem for imprimitivity systems. This implies in
particularthatifF isarationalfunctionwhichhasonlytwopolesandtheordersof
thesepolesaredifferentbetweenthemselvesthenanytwomaximaldecompositions
(2) of F have the same length and there exists a permutation σ ∈ S such that
r
the monodromy groups of U and V , 1≤i≤r, are permutation equivalent. We
i σ(i)
4 M.MUZYCHUK,F.PAKOVICH
also show that the Jordan-Ho¨lder theorem for imprimitivity systems holds for any
permutation group containing a transitive Hamiltonian subgroup that generalizes
the corresponding results of [21], [22].
For arbitrary rational functions the first Ritt theorem fails to be true. The
simplestcounterexamplesareprovidedbythefunctionswhichareregularcoverings
ofthe sphere (that is for which G =e)with the monodromygroupA , S , or A .
ω 4 4 5
These functions were described for the first time by F. Klein in [17] and nowadays
can be interpreted as Belyi functions of Platonic solids (see [6], [20]). For such
a function its maximal decompositions simply correspond to maximal chains of
subgroups in its monodromy group. Therefore, since any of the groups A , S , A
4 4 5
has maximal chains of subgroups of different length, for the corresponding Klein
functions the first Ritt theorem is not true.
Although the fact that the Klein functions provide counterexamples to the first
Ritt theorem is a well known part of the mathematical “folklore”, the systematic
description of compositional properties of these functions seems to be absent. In
particular,toourbestknowledgemaximaldecompositionswhichdonotsatisfythe
first Ritt theorem were found explicitly only for the Klein function corresponding
tothe groupA (see [15],[4]). Inthe Appendix to this paper weprovideadetailed
4
analysisofmaximaldecompositionsofthe Kleinfunctions andgiverelatedexplicit
examplesof nonweaklyequivalentmaximaldecompositions. In particular,we give
an example of a rationalfunction with three poles having maximal decompositions
of different length. This example shows that with no additional assumptions the
first Ritt theorem can not be extended to rational functions the monodromy of
which contains a cyclic subgroup with more than two orbits.
Acknowledgments. The authors would like to thank I. Kovacs, P. Mu¨ller, U.
Zannier,andA.Zvonkinfordiscussionsofdifferentquestionsrelatedtothesubject
of this paper.
2. Jordan-Ho¨lder theorem for imprimitivity systems
2.1. Lattices, imprimitivity systems, and decompositions of functions.
Recallthata latticeisapartiallyorderedset(L,≤)inwhicheverypairofelements
x,y has a unique supremum x∨y and an infimum x∧y (see e.g. [1]). Our basic
example of a lattice is a lattice L(G) of all subgroups of a group G, where by
definition G ≤ G if G is a subgroup of G (clearly, G ∩ G is an infimum of
1 2 1 2 1 2
G ,G and hG ,G i is a supremum). A simplest example of the lattice L(G) is
1 2 1 2
obtained if G is a cyclic group of order n. In this case L(G) is isomorphic to the
lattice L consisting of all divisors of n, where by definition d ≤d if d |d .
n 1 2 1 2
A sublattice of a lattice L is a non-empty subset M ⊆ L closed with respect to
∨ and ∧. For example, for any subgroup H of a group G the set
L(H,G):={X ≤G|H ≤X ≤G}
is a sublattice of L(G). Another example of a sublattice of L(G) is the lattice
L(A,AB):={X ≤G|A≤X ⊆AB}
(noticethatinournotationX ≤GmeansthatX isasubgroupofGwhileX ⊆AB
means that X is a subset of the set AB which in general is not supposed to be
a group). Recall that by the Dedekind identity (see e.g. [16], p. 8) for arbitrary
subgroupsA,B,X ofagroupGsuchthatA≤X ⊆AB theequalityX =A(X∩B)
JORDAN-HO¨LDER THEOREM FOR IMPRIMITIVITY SYSTEMS 5
holds. It follows from the Dedekind identity that the mapping f : X 7→ X ∩B is
a monomorphism from the lattice L(A,AB) into the lattice L(A∩B,B) with the
image consisting of all subgroups of B which are permutable with A. We will call
f the Dedekind monomorphism.
For elements a,b of a lattice L the symbol a<· b denotes that a≤b and there
existsnoelementc6=a,bofLsuchthata≤c≤b.AlatticeLiscalledsemimodular
[1] if for any a,b∈L the condition
(6) a∧b<· a, a∧b<· b,
imply the condition
(7) b<· a∨b, a<· a∨b.
If vice versa condition (7) implies condition (6), the lattice L is called lower semi-
modular. A lattice L is calledmodular ifL is semimodular andlowersemimodular.
A maximal chain R between elements a,b of L is a collection a ,a ,...a of ele-
0 2 k
ments of L such that
R: a=a <· a <· ... <· a =b.
0 1 k
The number k is called the length of the chain R (we always assume that in the
lattices considered the length of a chain between a and b is uniformly bounded by
a number depending on a and b only).
It is well known (see e.g. [1]) that for a semimodular or lower semimodular
lattice all maximal chains between two elements have the same length. Below,
using essentially the same proof, we give a modification of this statement in the
spirit of the first Ritt theorem.
Say that two maximal chains between elements a and b of a lattice L are r-
equivalent if there exists a sequence of maximal chains T ,T , ... T between a,b
1 2 s
such that T = R , T = R , and T is obtained from T , 1 ≤ i ≤ s−1, by a
1 1 s 2 i+1 i
replacement of exactly one element. Clearly, all r-equivalent chains have an equal
length.
Theorem 2.1. Let L be a semimodular or lower semimodular lattice. Then any
two maximal chains between any elements a and b of L are r-equivalent.
Proof. Since after the inversion of the ordering of a lattice the condition of semi-
modularity transforms to the condition of lower semi-modularity and vice versa,it
is enough to prove the theorem for lower semi-modular lattices.
Fix a∈L. Forarbitraryb∈Ldenotebyd(b)amaximumoflengthsofmaximal
chains between a and b. We will prove the theorem by induction on d(b). For b
satisfying d(b) ≤ 1 the theorem is obviously true. Suppose that the theorem is
proved for b satisfying d(b)≤n−1 and let
R : a=a <·a <· ... <·a =b, R : a=b <·b <· ... <·b =b
1 0 2 k1 2 0 2 k2
be two maximal chains between a and an element b∈L such that d(b)=n.
If ak1−1 = bk2−1, then we are done by induction. So, we may assume that
ak1−1 6= bk2−1. Then by the maximality of ak1−1 and bk2−1 in b we conclude
ak1−1∨bk2−1 =b. Hence
ak1−1 <· ak1−1∨bk2−1, bk2−1 <· ak1−1∨bk2−1
and therefore by the lower semi-modularity of L we have:
(8) ak1−1∧bk2−1 <· ak1−1, ak1−1∧bk2−1 <· bk2−1.
6 M.MUZYCHUK,F.PAKOVICH
Figure 1.
Let
a=c0 <· c2 <· ... <· cl =ak1−1∧bk2−1
be any maximal chain between a and ak1−1∧bk2−1 and
(9) a=c0 <· c2 <· ... <· cl <· ak1−1
be its extension to a maximal chain between a and ak1−1. Since d(ak1−1) is obvi-
ously less than d(b), it follows from the induction assumption that the chain
a=a0 <· a2 <· ... <· ak1−1
obtained from R by deleting a is r-equivalent to the chain (9). Therefore, the
1 k1
chain R and the chain
1
(10) a=c0 <· c2 <· ... <· cl <· ak1−1 <· b
also are r-equivalent.
Similarly, the chain R is r-equivalent to the chain
2
(11) a=c0 <· c2 <· ... <· cl <· bk2−1 <· b.
Since chains (10) and (11) are r-equivalent, we conclude that the chain R is r-
1
equivalent to the chain R . 2
2
Remark. Noticethatthereexistlatticeswhicharenotsemimodularorlowersemi-
modularsuchthat anytwo maximalchainsbetweenany elements arer-equivalent.
An example of such a lattice is shown on Fig. 1.
LetΩ be afinite setandG≤Sym(Ω)be atransitivepermutationgroup. Recall
that a partition E of Ω is called an imprimitivity system of G if E is G-invariant.
Elements of E are called blocks. For a point ω ∈ Ω we will denote by E(ω) a
unique block of E which contains ω. Since the group G permutes the elements of
E transitively, all blocks of E have the same cardinality denoted by nE. Denote by
E(G) the set of all imprimitivity systems of G. It is a partially ordered set, where
by definition E ≤ F if E is a refinement of F. Notice that if E ≤ F then nF/nE is
an integer denoted by [F :E].
It is easyto see thatE(G) is a lattice where the lattice operationsaredefined as
follows
E∧F :={∆∩Γ|∆∈E,Γ∈F and ∆∩Γ6=∅},
E∨F := {D∈E(G)|E≤D and F ≤D}.
^
JORDAN-HO¨LDER THEOREM FOR IMPRIMITIVITY SYSTEMS 7
ItiswellknownthatthelatticeE(G)isisomorphictothesubgrouplatticeL(G ,G)
ω
where ω ∈ Ω is an arbitrary fixed point. The correspondence between two sets is
given by the formula E7→GE(ω), where
GE(ω) :={g ∈G|E(ω)g =E(ω)}.
Vice versa, an imprimitivity system corresponding to a subgroup K ∈L(G ,G) is
ω
defined as follows E :={ωKg|g ∈G}. Notice that for any E,F ∈E(G) we have:
K
G(E∧F)(ω) =GE(ω)∩GF(ω), G(E∨F)(ω) =hGE(ω),GF(ω)i.
Moreover,if E≤F then [F :E]=[GF(ω) :GE(ω)].
IfagroupGisthemonodromygroupofarationalfunctionF,thenimprimitivity
systemsofGareinaone-to-onecorrespondencewithequivalenceclassesofdecom-
positions A◦B of F. Namely, suppose that G is realized as a permutation group
acting on the set z ,z ,...,z of preimages of a non critical value z of F =A◦B
1 2 n 0
under the map F : CP1 → CP1, and let x ,x ,...,x be the set of preimages of
1 2 r
z under the map A : CP1 → CP1. Then blocks of the imprimitivity system of G
0
correspondingtotheequivalenceclassofdecompositionsofF containingA◦B,are
just preimages of the points x ,x ,...,x under the map B : CP1 → CP1. More
1 2 r
generally, equivalence classes of decompositions of a rational function F are in a
one-to-one correspondence with chains of subgroups
G =T <T < ... <T =G,
ω 0 1 r
where G is the monodromy group of F.
Following [28] we say that two maximal decompositions D ,D of a rational
1 2
functionF areweaklyequivalentif there exists a chainof maximaldecompositions
F , 1 ≤ i ≤ s, of F such that F = D , F ∼ D , and F is obtained from F ,
i 1 1 s 2 i+1 i
1 ≤ i ≤ s−1, by replacing two successive functions in F by other functions with
i
thesamecomposition. Theremarksaboveimplythattwomaximaldecompositions
ofF areweaklyequivalentifandonlyifcorrespondingmaximalchainsinL(G ,G)
ω
arer-equivalent. Inparticular,the conclusionofthe firstRitttheoremis truefor a
rationalfunctionF ifandonlyifallmaximalchainsbetweenG andGinL(G ,G)
ω ω
are r-equivalent. Therefore, Theorem 2.1 implies the following corollary (cf. [28],
Th. 2.5).
Corollary 2.2. Let F be a rational function such that the lattice L(G ,G), where
ω
G is the monodromy group of F, is semi-modular or lower semi-modular. Then all
maximal decompositions of F are weakly equivalent. 2
The Corollary 2.2 shows that the groups G for which L(G ,G) is semi-modular
ω
or lower semi-modular are of special interest for factorization theory of rational
functions. The simplest examplesofsuchgroupsaregroupscontainingatransitive
cyclic subgroup.
Theorem 2.3. Let G ≤ S be a permutation group containing a transitive cyclic
n
subgroup C . Then the lattice L(G ,G) is a modular lattice isomorphic to a sub-
n 1
lattice of the lattice L .
n
Proof. Since any sublattice of a modular lattice is modular (see e.g. [1]) and it is
easy to see that L is modular, it is enough to prove that L(G ,G) is isomorphic
n 1
to a sublattice of L .
n
8 M.MUZYCHUK,F.PAKOVICH
ThetransitivityofC impliesthatG=G C . Therefore,theDedekindmonomor-
n 1 n
phismf :X 7→X∩C maps L(G ,G) into a sublattice ofL(G ∩C ,C ). Onthe
n 1 1 n n
other hand,
L(G ∩C ,C )=L(e,C )∼=L . 2
1 n n n n
Note that Theorem 2.3 implies the following proposition (cf. [9], [34]).
Corollary 2.4. Let A,B,C,D be polynomials such that
A◦C =B◦D.
Then there exist polynomials U,V,Aˆ,Cˆ,Bˆ,Dˆ, where
degU =gcd(degA,degB), degV =gcd(degC,degD),
such that
A=U ◦Aˆ, B =U ◦Bˆ, C =Cˆ◦V, D =Dˆ ◦V,
and
Aˆ◦Cˆ =Bˆ◦Dˆ.
In particular, if degA = degB then the decompositions A ◦ C and B ◦ D are
necessarily equivalent.
2.2. Jordan-H¨older theorem for groups with normal imprimitivity sys-
tems. Let as above G be a transitive permutation group. It is easy to see that
if N is a normal subgroup of G then its orbits form an imprimitivity system of
G. Such an imprimitivity system is called normal and is denoted by Ω/N. For an
imprimitivity system E∈E(G) set
GE :={g ∈G|∀∆∈E ∆g =∆}.
Notice that each block of E is a union of GE-orbits and GE = coreG(GE(ω)). In
particular, GE is a normal subgroup of G.
Let us call a subgroup A∈L(G ,G) core-complementary if A=G core (A).
ω ω G
Proposition 2.5. An imprimitivity system E ∈ E(G) is normal if and only if the
group GE(ω) is core-complementary.
Proof. Indeed, if
(12) GE(ω) =GωcoreG(GE(ω))=GωGE
then
E(ω)=ωGE(ω) =ωGE
and hence GE acts transitively on E(ω). Since GE E G, this implies that GE acts
transitivelyoneveryblockofE. ThusblocksofEareorbitsofthenormalsubgroup
GE.
Vice versa, if E is normal then E = Ω/N for some N E G. This implies that
GE(ω) =GωN and N ≤GE. It follows now from
GE(ω) =GωN ≤GωGE ≤GE(ω)
that equality (12) holds. 2
Recall that two subgroups A and B are called permutable if AB = BA, or,
equivalently, hA,Bi = AB. Recall also that if A and B are subgroups of finite
index of G then the inequality
(13) [hA,Bi:B]≥[A:A∩B]
JORDAN-HO¨LDER THEOREM FOR IMPRIMITIVITY SYSTEMS 9
holds and the equality in (13) attains if and only if A,B are permutable (see e.g.
[19], p. 79).
Denote by L (G ,G) the subset of L(G ,G) consisting of all core-complemen-
c ω ω
tary subgroups. Notice that in general L (G ,G) is not a sublattice of L(G ,G)
c ω ω
Proposition 2.6. The following conditions hold:
(a) If A∈L (G ,G), then AB =BA for each B ∈L(G ,G);
c ω ω
(b) If A,B ∈L (G ,G), then AB ∈L (G ,G).
c ω c ω
Proof. (a)InordertolightenthenotationsetN =core (A). InviewofProposition
G
2.5 we have:
AB =G NB =NG B =NB =BN =BG N =BA.
ω ω ω
(b) Set M =core (B). Since MN EG and MN ≤AB, we have:
G
MN ≤core (AB).
G
It follows now from Proposition 2.5 that
AB =G NG M =G MN ≤G core (AB)≤AB.
ω ω ω ω G
Therefore, G core (AB) = AB and hence AB ∈ L (G ,G) by Proposition 2.5.
ω G c ω
2
Proposition 2.7. Let A,B ≤G be permutable subgroups. If A∩B is maximal in
A and B, then A and B are maximal in hA,Bi=AB.
Proof. Let A be a subgroup of G satisfying A≤A ≤AB. It follows from
1 1
A∩B ≤A ∩B ≤B
1
that either A ∩B = A∩B or A ∩B = B. It follows now from the Dedekind
1 1
identity A = A(A ∩B) that in the first case A = A while in the second one
1 1 1
A =AB. 2
1
Proposition2.8. IfanytwosubgroupsofL(G ,G)arepermutable,thenthelattice
ω
L(G ,G) is modular.
ω
Proof. Indeed, if A∩B is maximal in A and B then A and B are maximal in
hA,Bi=AB by Proposition2.7.
Suppose now that A and B are maximal in AB and let A be a subgroup of G
1
satisfying A∩B ≤A ≤A. Then
1
B ≤A B ≤AB
1
implies that either B = A B or A B = AB. If B = A B, then A ≤ B and
1 1 1 1
therefore A = A∩ B. On the other hand, if A B = AB then it follows from
1 1
A≤AB =A Bthatforanya∈Athereexista ∈A andb∈Bsuchthata=a b.
1 1 1 1
Sincethe lastequalityyieldsthatb∈A∩B,this impliesthatA≤A (A∩B)≤A
1 1
and hence A =A. 2
1
Let H ≤ G be an arbitrary subgroup and H\G := {Hx|x ∈ G}. Denote by
G//H a permutation group arising from the natural action of G on H\G. Thus
G//H is always considered as a subgroup of Sym(H\G). Notice that if N E G
is contained in H, then the groups G//H and (G/N)//(H/N) are permutation
equivalent. Below we will denote permutation equivalence by ∼= .
p
10 M.MUZYCHUK,F.PAKOVICH
SaythatatransitivepermutationgroupG≤Sym(Ω)satisfiestheJordan-Ho¨lder
theorem for imprimitivity systems if any two maximal chains
G =A <...<A =G and G =B <...<B =G
ω 0 k ω 0 m
ofthelatticeL(G ,G)havethesamelength(thatisk =m)andthereexistsaper-
ω
mutation σ ∈ Sk such that the permutation groups Ai//Ai−1 and Bσ(i)//Bσ(i)−1,
1≤i≤k,arepermutationequivalent. Notice thatifGisthemonodromygroupof
arationalfunctionF thenitfollowsfromthecorrespondencebetweenimprimitivity
systems of G and equivalence classes of decompositions of F that G satisfies the
Jordan-Ho¨lder theorem for imprimitivity systems if and only if any two maximal
decompositions of F
F =U ◦U ◦···◦U and F =V ◦V ◦···◦V ,
1 2 k 1 2 m
have the same length and the there exists a permutation σ ∈ S such that the
k
monodromy groups of U and V , 1≤i≤k, are permutation equivalent.
i σ(i)
Theorem 2.9. Let G be a permutation group such that L(G ,G) = L (G ,G).
ω c ω
Then the lattice L(G ,G) is modular and G satisfies the Jordan-Ho¨lder theorem
ω
for imprimitivity systems.
Proof. First of all observe that since by Proposition 2.6 any two subgroups of
L(G ,G)arepermutableitfollowsfromProposition2.8thatL(G ,G)isamodular
ω ω
lattice. Let now
A:=G =A <...<A =G, and B:=G =B <...<B =G
ω 0 k ω 0 m
be twomaximalchainsofL(G ,G). Since L(G ,G)is a modularlattice, itfollows
ω ω
fromTheorem2.1thatk =mandAandBarer-equivalent. Thereforebyinduction
it is sufficient to prove the theorem for the case when B and A differs at exactly
one place, say i (1≤i<k). Clearly, in this case we have:
Ai−1 =Bi−1 =Ai∩Bi, Ai+1 =Bi+1 =AiBi.
In order to lighten the notation set
N :=core (A ).
Ai+1 i
ItfollowsfromtheequalityAi =GωcoreG(Ai)thatAi =Ai−1coreG(Ai). Therefore,
Ai =Ai−1coreG(Ai)≤Ai−1N ≤Ai
and hence
Ai =Ai−1N =Bi−1N
and
Ai+1 =AiBi =Ai−1NBi =Bi−1NBi =BiN.
Since N ≤Ai =Bi−1N and N EAi+1 =BiN, this implies that
Ai+1//Ai =(BiN)//(Bi−1N)∼=p (BiN)/N//(Bi−1N)/N.
By the Second Isomorphism Theorem the group (B N)/N is isomorphic to the
i
group Bi/(Bi∩N) and the image of (Bi−1N)/N under this isomorphism is
Bi−1(Bi∩N)/(Bi∩N).
Furthermore, it follows from N ≤Ai that Bi∩N ≤Ai∩Bi =Bi−1. Therefore,
Bi−1(Bi∩N)/(Bi∩N)=Bi−1/(Bi∩N)