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## Mathematik.uni-wuerzburg.de

Contemporary Mathematics

Volume

**00**, 0000

**Primitive Monodromy Groups of Polynomials**
Abstract. For a polynomial

*f ∈ *C[

*X*], let

*G *be the Galois group of theGalois closure of the field extension C(

*X*)

*|*C(

*f *(

*X*)). We classify the groups

*G *in the indecomposable case. For polynomials with rational coefficientsthere are, besides four infinite series, only three more “sporadic” exam-ples. In the Appendix we reprove the classical Theorems of Ritt aboutdecompositions of polynomials using the group-theoretic setup.

**1. Introduction**
Let

*f *be a polynomial of degree

*n *with complex coefficients. In a fixed alge-
braic closure of the field of rational functions C(

*t*) consider the field Ω which isgenerated over C by the

*n *different elements

*xi *fulfilling

*f *(

*xi*) =

*t*. Then theGalois group

*G *= Gal(Ω

*|*C(

*t*)) permutes transitively the elements

*xi*. This group

*G *is called the monodromy group of

*f *. It is natural to ask what groups

*G *canoccur this way. A polynomial is called

*indecomposable *if it cannot be written as acomposition of two non–linear polynomials. In section 2 we classify the possiblemonodromy groups for indecomposable polynomials, there are four infinite seriesand twelve more cases which do not belong to these series. Section 3 is aboutthe question of what groups occur as monodromy groups of polynomials withrational coefficients. The result is
Theorem.

*Let f ∈ *Q[

*X*]

*be indecomposable and let G be its monodromy*
*Then G is either alternating, symmetric, cyclic, or dihedral or G is*
*PGL*2(5)

*, P*Γ

*L*2(8)

*, or P*Γ

*L*2(9)

*. In the latter three cases f is, up to compositionwith linear polynomials, uniquely given by X*4(

*X*2 + 6

*X *+ 25)

*, *9

*X*9 + 108

*X*7 +72

*X*6+486

*X*5+504

*X*4+1228

*X*3+888

*X*2+1369

*X or *(

*X*2

*−*405)4(

*X*2+50

*X*+945)

*.*
Some remarks about the appearance of monodromy groups are in order: Es-
pecially M. Fried (see [

**7**], [

**8**], [

**9**], and his papers cited in [

**11**]) exhibited the

1991

*Mathematics Subject Classification*. Primary 11C08, 20B15, 20B20; Secondary 12E05.

Financial support from the DAAD is greatfully acknowledged.

This paper is in final form, and no version of it will be submitted for publication elsewhere.

importance of these groups in discussing several arithmetical questions about

polynomials. That is many questions depend merely on the monodromy group

rather than on the full information given by a polynomial. One of these problems

is a question of Davenport, which asks to classify the pairs of polynomials with

integer coefficients such that the value sets on Z are the same modulo all but

finitely many primes. See [

**11, 19.6**], [

**9**], and [

**20**].

We merely classify the monodromy groups in the indecomposable case. By
a Theorem of Ritt (see [

**21**] or [

**3**]) the study of arbitrary polynomials can be

reduced to these polynomials to some extent. For instance (over fields of char-

acteristic 0) any two decompositions of a polynomial into indecomposable poly-

nomials have the same number of factors, and the degrees are the same up to

a permutation. Ritt even gives an algorithm how to pass from one composi-

tion to the other one by interchanging and “twisting” consecutive factors. In

the Appendix, we give a concise account of this, employing the group-theoretic

setup.

olklein for drawing my attention to this question. I
thank B. H. Matzat for informing me about [

**17**] where he already computed

the polynomials for the groups PΓL2(8) and PΓL2(9). He also noted that I

erroneously excluded PΓL2(8) in an earlier version of this paper.

**2. Primitive Monodromy Groups**
**2.1 Notation and Definitions. **We retain the notation from the Introduc-

tion. For technical reasons we need a further description of the monodromygroup of

*f ∈ *C[

*X*] (deg

*f *=

*n*):
Consider the branched

*n*-fold covering

*f *: P1

*→ *P1. Let

*S *=

*{p*1

*, p*2

*, . . . , pr}*
be the set of branch points, where

*pr *is the point at infinity. Fix a point

*a *inP1

*\ S*. Then

*π*1 =

*π*1(P1

*\ S, a*) acts on

*f−*1(

*a*) by lifting of paths. The homo-morphic image of

*π*
1 in S

*n *= Sym(

*f −*1(

*a*)) will also be denoted by

*G*, as this
group can be identified with the monodromy group defined in the Introduction,with

*G *acting in the same way on the elements

*xi *as on the points of the fiber

*f −*1(

*a*). This identification relies on the isomorphism between the group of cov-ering transformations of a Galois cover of compact connected Riemann surfacesand the Galois group of the corresponding extension of fields of meromorphicfunctions on these surfaces.

In this section we use the geometric description of

*G *from above, i.e. we view

*G *as a subgroup of S

*n *via identification of

*{*1

*, *2

*, . . . , n} *with

*f −*1(

*a*).

Pick

*r *generators

*λi *of

*π*1(P1

*\ S, a*) such that

*λi *winds only around

*pi *and

*λ*1

*λ*2

*· · · λr *= 1 (this is a so–called “standard homotopy basis”). These

*r *gener-ators of

*π*1(P1

*\ S, a*) then yield generators

*σ*1

*, σ*2

*, . . . , σr *of

*G *with

*σ*1

*σ*2

*· · · σr *= 1

*.*
As

*pr *=

*∞ *ramifies completely,

*σr *is an

*n*-cycle.

PRIMITIVE MONODROMY GROUPS OF POLYNOMIALS
This tuple (

*σ*1

*, . . . , σr*) is called the branch cycle description of the cover
For

*σ ∈ *S

*n *denote by ind

*σ *the quantity ‘

*n − *the number of orbits of

*σ *’.

The main constraint is imposed by the Riemann Hurwitz genus formula
ind

*σi *= 2(

*n − *1)

*.*
For an elementary argument yielding this latter relation confer [

**7, Lemma 5**].

Conversely, a finite permutation group having a set of generators fulfilling the

above restrictions is the monodromy group of a suitable polynomial by Riemann’s

existence Theorem.

So we are reduced to a completely group–theoretic question. The purpose of
this section is to give a complete classification in the indecomposable case. The

corresponding question for rational functions instead of polynomials is much

tougher and still open, see [

**15**] and [

**1**].

**2.2 Notations and Main Result. **Let

*G *be a permutation group acting

on

*n *elements. We consider the following condition on

*G*, which we later referto as (*).

Condition (*).

*G *is generated by

*σ*1

*, σ*2

*, . . . , σs *(

*σi *= 1) such that

*σ*1

*σ*2

*· · · σs*
It is obvious that the situation in (*) is equivalent to the configuration of 2.1.

Suppose (*) holds. Using

*ab *=

*bab *we see that we may assume

*|σ*1

*| ≤ |σ*2

*| ≤*
*· · · ≤ |σs|*, where

*|σ| *denotes the order of

*σ*. Following Feit in [

**4**] we say that

*G*

is of type (

*|σ*1

*|, |σ*2

*|, · · · , |σs| *:

*n*).

Denote by

*Cp *and

*Dp *the cyclic and dihedral groups of degree

*p*, respectively.

Let PGL

*k*(

*q*) be the projective linear group over the field with

*q *elements, act-ing on the projective space of dimension

*k − *1. This group, together with thecomponent–wise action of Aut(F

*q*) on the projective space, generates the semi-linear group PΓL

*k*(

*q*). The Mathieu groups of degree

*n *are labelled by M

*n*. Inthis section we prove
Theorem.

*Let G be the monodromy group of a polynomial f ∈ *C[

*X*]

*. Then*
*G is one of the following groups. Conversely, each of these groups occurs.*
(i)

*Cp of type *(

*p *:

*p*)

*, p a prime.*
*Dp of type *(2

*, *2 :

*p*)

*, p an odd prime.*
(ii)

*PSL*2(11)

*of type *(2

*, *3 : 11)

*PGL*3(2)

*of types *(2

*, *3 : 7)

*, *(2

*, *4 : 7)

*, and *(2

*, *2

*, *2 : 7)

*PGL*3(3)

*of types *(2

*, *3 : 13)

*, *(2

*, *4 : 13)

*, *(2

*, *6

*, *13)

*, and *(2

*, *2

*, *2 : 13)

*PGL*4(2)

*of types *(2

*, *4 : 15)

*, *(2

*, *6 : 15)

*, and *(2

*, *2

*, *2 : 15)

*P*Γ

*L*3(4)

*of type *(2

*, *4 : 21)

*PGL*5(2)

*of type *(2

*, *4 : 31)
(iii)

*An (n odd) and Sn of many, not reasonably classifiable types.*
*M*11

*of type *(2

*, *4 : 11)

*M*23

*of type *(2

*, *4 : 23)

*PGL*2(5)

*of type *(2

*, *4 : 6)

*PGL*2(7)

*of type *(2

*, *3 : 8)

*P*Γ

*L*2(8)

*of types *(2

*, *3 : 9)

*and *(3

*, *3 : 9)

*P*Γ

*L*2(9)

*of type *(2

*, *4 : 10)
Remark. In [

**19, 2.6.10**] we get, as a side product to the Hilbert–Siegel prob-

lem, a classification of the monodromy groups of the rational functions

*f *(

*X*)

*/X*where

*f *is an arbitrary polynomial

*f ∈ *C[

*X*] with

*f *(0) = 0. The list is asfollows: AGL1(

*p*) with

*p ∈ {*2

*, *3

*, *5

*, *7

*}*, AΓL1(8), AGL3(2), AΓL2(4), AGL4(2),AGL5(2), AΓL1(9), AGL2(3), A

*n *(

*n *even), S

*n*, PSL2(5), PGL2(5), PSL2(7),PGL2(7), PSL2(13), M11 with

*n *= 12, M12, M24.

There are also results about monodromy groups of indecomposable polynomi-
als with coefficients in a finite field or in an algebraically closed field of positive

characteristic. In [

**14**] is a classification of the primitive groups which meet a

necessary condition for being the monodromy group of a polynomial. The main

constraint comes from the ramification at infinity. The genus condition however

is hardly to use.

**2.3 About the Proof. **Let

*f ∈ *C[

*X*] be a polynomial and

*G *its monodromy

uroth’s Theorem,

*f *is indecomposable if and only
if

*G *is a primitive group (i.e.

*G *does not act on a non–trivial partition of the

underlying set), see [

**12, 3.4**].

Now let

*f *be indecomposable and

*σ*1

*, . . . , σs *a generating system of

*G *fulfilling
(*). Set

*Z *=

*σ*1

*σ*2

*· · · σs *, then

*Z *is a transitive cyclic subgroup of

*G*. Together

with the primitivity of

*G*, we get by classical results of Schur and Burnside (see

[

**22, 11.7 and 25.2**]) that

*G ≤ *AGL1(

*p*) (

*p *a prime) or

*G *is doubly transitive.

Let

*A *be a minimal normal subgroup of

*G*. If

*A *is elementary abelian, then

*G ≤ *AGL1(

*p*) or

*G ≤ *S4, see the proof of [

**16, Satz 5**]. The non–solvable doubly

transitive groups with a cyclic transitive subgroup are known by the classification

of the finite simple groups and listed in [

**5, 4.1**]. Thus we have to investigate

the following groups

*G*.

*Cp ≤ G ≤ *AGL1(

*p*), A

*n *(

*n *odd), S

*n*, M11, M23, PSL2(11) of degree11, and (not all) groups between PSL

*m*(

*q*) and PΓL

*m*(

*q*) (with

*m ≥ *2,

*q *a prime power) in its action on the projective space.

The cases

*G *= PSL2(11) with

*n *= 11 and the semi–projective linear groups
with

*m ≥ *3 have been investigated by Feit in [

**4**]. (These are just the cases

where

*G *admits two inequivalent doubly transitive representations with

*Z *acting

transitively in both of them.) However, his proof needs to be modified, as [

**4,**

3.4] is wrong. In the following we supply an alternative treatment in the case

when [

**4, 3.4**] does not work. I thank Feit for a discussion about this.

PRIMITIVE MONODROMY GROUPS OF POLYNOMIALS
We will discuss the different classes of groups separately. The case

*Cp ≤ G ≤*
AGL1(

*p*) is quite easy and left to the reader.

**2.4 Counting Orbits. **Some more notation: For

*σ ∈ G *denote by

*o*(

*σ*)

the number of cycles of

*σ *(thus

*n *=

*o*(

*σ*) + ind

*σ*). Let

*f *(

*σ*) be the number ofelements fixed by

*σ*. For later use we derive an elementary relation between

*o*(

*σ*)and the number of fixed points of powers of

*σ*: For this denote by

*ai *the numberof

*i*-cycles of

*σ*. Clearly

*µ*( )

*f *(

*σk*)

*.*
*µ*(

*t*) =

*ϕ*(

*m*) and

*o*(

*σ*) =

*a*
1 +

*a*2 +

*. . . *we get the basic relation

**2.5 The Mathieu Groups. **For the two candidates

*M*11 and

*M*23 we use

the Atlas of the finite simple groups [

**2**] and its notation. Besides other things

the character tables in this source allow us to compute the ind–function. Let

us start with

*G *=

*M*11: First we get ind

*σi ≥ *4, hence

*s *= 2 by (*). We see

that

*σ*1

*∈ *2

*A *and

*σ*2

*∈ {*3

*A, *4

*A}*. Let

*χ*1,

*χ*2,

*. . . *,

*χh *be the irreducible characters

of

*G *and

*C*1,

*C*2,

*. . . *,

*Ch *be the conjugacy classes of

*G*. For an

*x ∈ Ck *denote

by

*N *(

*i, j*;

*k*) the number of solutions of

*x *=

*uv *with

*u ∈ Ci *and

*v ∈ Cj*. It is

well–known (see e.g. [

**13, 4.2.12**]) that

*χm*(

*Ci*)

*χm*(

*Cj*)

*χm*(

*Ck*)
We want to exclude the case

*σ*2

*∈ *3

*A*: Pick an element

*g ∈ G *of order 11. Using

(2) we see that there are exactly 11 solutions of

*g *=

*uv *with

*u ∈ *2

*A*,

*v ∈ *3

*A*.

Since

*g *is transitive and abelian

*g *does not centralize

*u *(and

*v*). Thus

*g *acts

fixed–point–freely on the pairs (

*u, v*) with

*g *=

*uv*. So there is essentially one

solution to

*g *=

*uv *with

*u ∈ *2

*A *and

*v ∈ *3

*A*. Now [

**2**] tells us that

*G *contains

a transitive subgroup isomorphic to

*H ∼*
= PSL2(11). Again using (2) and [

**2**] we

see that there are elements

*g*,

*u*, and

*v *in

*H *of orders 11, 2, and 3 respectivelywith

*g *=

*uv*. The previous consideration shows that a conjugate of

*σ*1

*, σ*2 is asubgroup of

*H*, therefore

*σ*1 and

*σ*2 do not generate

*G*.

Now consider the case

*σ*2

*∈ *4

*A*. Set

*H *=

*σ*1

*, σ*2 . If

*H *=

*G *then

*H ≤*
PSL2(11), as PSL2(11) is the only maximal and transitive subgroup of M11 by

[

**2**]. However, PSL2(11) does not contain an element of order 4. Thus

*σ*1 and

*σ*2

generate

*G*. An explicit example is

*σ*1 = (4

*, *5)(6

*, *7)(8

*, *9)(11

*, *11),

*σ*2 = (1

*, *11

*, *2

*, *9)(8

*, *3

*, *5

*, *7).

We treat the case

*G *=

*M*23 quite similarly: Here we get

*σ*1

*∈ *2

*A*,

*σ*2

*∈ *4

*A*, and

*H *=

*σ*1

*, σ*2 with

*σ*1

*σ*2 an 23-cycle. The only transitive and maximal subgroup

of

*G *has order 253, see [

**2**]. Thus

*σ*1 and

*σ*2 generate

*G*. Again we give one

explicit example:

*σ*1 = (8

*, *9)(10

*, *11)(12

*, *13)(14

*, *15)(16

*, *17)(18

*, *19)(20

*, *21)(22

*, *23)

*,*

σ2 = (2

*, *14)(17

*, *19)(1

*, *10

*, *16

*, *4)(8

*, *13

*, *15

*, *6)(12

*, *20

*, *11

*, *7)(3

*, *9

*, *5

*, *22)

**2.6 PSL**2(

*q*)

*≤ G ≤ ***PGL**2(

*q*)

**. **The distinction between the projective case

and the non–projective semilinear case simplifies the somewhat tedious waythrough the estimations. We assume

*q ≥ *5 (because PGL2(4)

*∼*
For the case

*q *= 11 and

*G *of degree 11 see [

**4, 4.3**]. Thus assume from now

on that

*G *acts naturally on the projective line.

Pick a

*σ ∈ G*. There are three cases:
(i)

*σ *has at least 2 fixed points. Then ind

*σ *= (

*q − *1)(1

*− *1
(ii)

*σ *has exactly one fixed point. Then ind

*σ *=

*q*(1

*− *1

*|σ| *) and

*|σ| *=

*p*.

(iii)

*σ *has no fixed points. Denote by ˆ

*σ *a preimage of

*σ *in GL2(

*q*). Then ˆ

*σ*] is a quadratic field extension of F

*q *by
Schur’s Lemma. We deduce that

*σ *acts fixed–point–freely on

**P**1(

*q*),

thus ind

*σ *= (

*q *+ 1)(1

*− *1

In all three cases we obtain ind

*σ ≥ *(

*q − *1)(1

*− *1

*s *= 2 by (*). As

*G *contains the non–solvable group PSL2(

*q*),

*σ*2 cannot be aninvolution (recall the monotony of the orders of

*σi*). This shows (again using(*)) (

*q − *1)(1

*− *1 + 1

*− *1 )

*≤ q*, hence

*q ≤ *7.

Suppose

*q *= 5. We have ind

*σ ∈ {*2

*, *3

*}*, ind

*σ *= 4, or ind

*σ ∈ {*3

*, *4

*} *if

*σ *is of
type (i), type (ii), or type (iii) respectively. Thus both

*σ*1 and

*σ*2 are of type (i)with

*|σ*1

*| *= 2,

*|σ*2

*| *= 4. One readily checks (see the arguments in the Mathieugroup case) that

*σ*1 and

*σ*2 generate a group containing PSL2(5). Since

*σ*2 isan odd permutation, they actually generate PGL2(5). An explicit example is

*σ*1 = (1

*, *2)(3

*, *4),

*σ*2 = (3

*, *2

*, *5

*, *6).

Now suppose

*q *= 7. Similarly as above we get

*|σ*1

*| *= 2,

*|σ*2

*| *= 3, and

*f *(

*σ*1) =

*f *(

*σ*2) = 2. So

*σ*1 is an odd permutation, hence

*G *= PGL2(7). This case occursas well, an example is provided by

*σ*1 = (1

*, *2)(3

*, *4)(5

*, *6),

*σ*2 = (1

*, *3

*, *6)(2

*, *7

*, *8).

**2.7 PSL***m*(

*q*)

*≤ G ≤ ***P**Γ

**L***m*(

*q*)

**. **Set

*q *=

*pe *with a prime

*p*. This case is even

for

*m *= 2 more complicated than the projective linear case since there are manymore types of possible cycle decompositions. In particular a fixed–point–freeelement need not generate a fixed–point–free group.

Write ΓL

*m*(

*q*) = GL

*m*(

*q*)
Γ with Γ = Aut(F

*q*). Let ΓL

*m*(

*q*) act from the left

*σ *a preimage of

*σ *in ΓL

*m*(

*q*). Feit [

**6**] told

me a special case of the following lemma.

*σ *=

*gγ with g ∈ GLm*(

*q*)

*and γ ∈ *Γ

*. Let e/i be the order of*
PRIMITIVE MONODROMY GROUPS OF POLYNOMIALS

*γ. Then f *(

*σ*)

*≤ pim−*1

*.*
*σe/i *=

*h ∈ *GL

*m*(

*q*). If ˆ

*σv *=

*αv *for

*v ∈ *F

*m \ {*0

*}*,

*α ∈ *F

*hv *=

*N *(

*α*)

*v*, where

*N *: F

*q → *F

*pi *denotes the norm.

First suppose that

*h *is a scalar. If ˆ

*σw *=

*βw *for some

*w ∈ *F

*m \ {*0

*}*,

*β ∈ *F
then

*hw *=

*N *(

*β*)

*w *and therefore

*N *(

*α*) =

*N *(

*β*). By Hilbert’s Theorem 90 thereis a

*ζ ∈ *F

*× *with

*α/β *=

*ζγ/ζ*. From ˆ

*σw *=

*ζγ βw *=

*αζw *we conclude

*σ*–invariant F

*q*–line

*L *contains an element

*u *= 0 with ˆ
are exactly

*pi − *1 such points on

*L*. One easily sees that a basis of the F

*pi*–space

*{v ∈ *F

*m| *ˆ

*σv *=

*αv} *is linearly independent over F

*q *. This proves the assertion.

Now suppose that

*h *is not a scalar.

Let

*η*1

*, . . . , ηs *be those elements in
F

*pi *which are eigenvalues of

*h*. Let

*Vk *be the eigenspace of

*h *with eigenvalue

*ηk*. Note that

*h|V *is the smallest power of ˆ

*σ*–invariant line lies in one of these subspaces. The
From (

*xδ*1

*− *1) + (

*xδ*2

*− *1)

*≤ xδ*1+

*δ*2

*− *1 for

*x ≥ *1 and

*δ*1

*, δ*2

*≥ *0 and

*σ *=

*gγ with g ∈ GLm*(

*q*)

*and γ ∈ *Γ

*. Let e/i be the order*
*of γ. Set r *= 1

*if e *=

*i. Otherwise let r be the smallest prime divisor of e/i.*

Then
*σ| *)(

*qm−*1

*− *1) + (1

*− *1
Proof. We use the formula in 2.4, together with the well–known relation

*ϕ*(

*t*) =

*m*. If

*σk /*
*m*(

*q*), then we estimate the number of fixed points
with the preceding lemma. Note that

*γk *has the order
(

*a, b*) denotes the greatest common divisor of

*a *and

*b*. If however 1 =

*σk ∈*PGL

*m*(

*q*), then clearly

*f *(

*σ*)

*≤ *1 (

*qm−*1

*− *1 +

*q − *1) =

*qm−*1

*−*1 + 1

*.*
*− qm−*1

*− *1

*− *1)

*ϕ*(1)+
+ 1

*− qm/r − *1 )

*· ϕ*(
+ 1

*− qm/r − *1 )

*· ϕ*(

*t*) +
If

*i < e*, then we get, using 2

*≤ r ≤ e/i ≤ |σ|*,
Corollary.

*Let σ ∈ P*Γ

*Lm*(

*q*)

*\ PGLm*(

*q*)

*.*

If m ≥ 4

*, then*
*σ| *)(

*qm−*1

*− *1) + 2

*If m ∈ {*2

*, *3

*}, then*
*|σ|*)(

*qm−*1

*− q*1

*/*2)

*.*
Now we are prepared to discuss condition (*). Let

*σ , . . . , σ ∈ G ≤ *PΓL
If not all the

*σ *are involutions, then assume without loss that

*σ *is not an
involution, and set

*σ*1 =

*σ σ . . . σ*
If all the

*σ *are involutions, then

*s ≥ *3, as

*G *is not dihedral. By conjugation
and operations of the kind

*. . . , a, b, . . . → . . . , b, ab, . . . *we may assume that
PRIMITIVE MONODROMY GROUPS OF POLYNOMIALS
and

*σ *do not commute. Then

*σ*
*σ *is not an involution, and we set
In either case we have

*σ*1

*, σ*2

*∈ G *which not both are involutions, such that
ind

*σ*1+ind

*σ*2

*≤ n−*1 (with

*n *= (

*qm−*1)

*/*(

*q−*1)) and such that

*σ*1

*σ*2 is an

*n*–cycle.

(As to the inequality for the index note that ind

*σ *is also the minimal number oftranspostions required to write

*σ *as a product with. Thus ind

*στ ≤ *ind

*σ *+ind

*τ *.

From this we actually get ind

*σ*1 + ind

*σ*2 =

*n − *1.)
If

*σ*1

*, σ*2

*∈ *PGL2(

*q*), then

*q *= 5 or 7 as in section 2.6. If

*σ*1

*, σ*2

*∈ *PGL

*m*(

*q*)
for

*m ≥ *3, then proceed as in [

**4**]. The key tool [

**4, 3.4**] is correct in this case.

From now on suppose that one of the elements

*σ*1

*, σ*2 is not contained in
PGL

*m*(

*q*). As a consequence of Zsigmondy’s Theorem and Schur’s Lemma, we

get that the

*n*–cycle

*σ*1

*σ*2 is contained in PGL

*m*(

*q*) except possibly for

*m *= 2

*, q *=

8, see the proof of [

**4, 5.1**]. The case

*m *= 2

*, q *= 8 is excluded until otherwise

stated.

Thus

*σ*1 and

*σ*2 have the same order

*e/i ≥ *2 modulo PGL

*m*(

*q*). In particular

*|σ*1

*| *and

*|σ*2

*| *have a common divisor

*> *1.

First suppose

*m ≥ *4. Using

*n − *1

*≥ *ind

*σ*1 + ind

*σ*2 and the Corollary we get

*≥ *((1

*− *1) + (1

*− *1))(

*qm−*1

*− *1) + (

*− qm/*2

*− *1 + 1)

*.*
As the last summand on the right side is positive, we get

*≥ *5(

*qm−*1

*− *1)

*,*
hence

*q *= 4. Now we use

*q *= 4 in (2), and easily get the contradiction

*m ≤ *3.

Now suppose

*m *= 3. Similarly as above we get

*q*(

*q *+ 1)

*≥ *5 (

*q*2

*− q*1

*/*2)

*,*
hence

*q *= 4. For a treatment of

*G ≤ *PΓL3(4) confer [

**4**].

From now on suppose

*m *= 2. Without loss we assume

*|σ*1

*| ≤ |σ*2

*|*. As above

*q ≥ *5 (

*q − q*1

*/*2)

*,*
If

*q *= 25, then

*|σ*1

*| *= 2,

*|σ*2

*| *= 4, and

*σ*2

*∈ *PGL
2(25), hence

*f *(

*σ*2)

*≤ f *(

*σ*2
2. From 2.4 we get ind

*σ*2

*≥ *18, hence ind

*σ*1

*≤ *7, contrary to the Corollary.

Now suppose

*q *= 16. We quickly get

*|σ*1

*| *= 2 and

*|σ*2

*| *= 4. As

*σ*1 and

*σ*2 have
the same order modulo PGL2(16), we get

*σ*2

*∈ *PGL
one fixed point, hence so does

*σ*2. It follows ind

*σ*2 = 12, hence ind

*σ*1 = 4,contrary to the Corollary.

Now suppose

*q *= 9. We get

*s *= 2, for if

*s ≥ *3 then

*s *= 3 and the

*σi *are
fixed–point–free involutions. However PGL2(9) does not contain fixed–point–freeinvolutions, contrary to

*σ*1

*σ*2

*σ*3

*∈ *PGL2(9).

So we get

*|σ*1

*| *= 2 and

*|σ*2

*| *= 4. An explicit example is

*σ*1 = (2

*, *7)(5

*, *6)(8

*, *10),

*σ*2 = (1

*, *4

*, *9

*, *2)(3

*, *5

*, *7

*, *10).

For the last case

*G *= PΓL2(8) we get from the index estimations

*s *= 2 and
(

*|σ*1

*|, |σ*2

*|*) = (2

*, *3) or (3

*, *3). Explicit examples are

*σ*1 = (2

*, *3)(4

*, *5)(6

*, *7)(8

*, *9),

*σ*2 = (7

*, *5

*, *8)(1

*, *9

*, *3) and

*σ*1 = (4

*, *5

*, *6)(7

*, *8

*, *9),

*σ*2 = (5

*, *9

*, *2)(6

*, *3

*, *1).

= S5 we do not discuss

*q *= 4.

**3. Rationality Questions**
**3.1. **Using a special case of the so–called branch cycle argument (for a short

proof see [

**10**]), we get the following

Lemma.

*Let f ∈ *Q[

*X*]

*be a polynomial of degree n. Let G be the monodromy*
*group of f , and let σ be an n–cycle as in (*). Let *ˆ

*Sn. Then any two generators of Z *=

*σ are conjugate in *ˆ
We now prove the Theorem from the Introduction.

We note that

*f *is indecomposable even over C by [

**12, 3.5**]. Thus we apply

our result from section 2.2. As for type (i), consider the polynomials

*f *(

*x*) =

*xp*or

*f ∈ *Q[

*X*] defined by

*f *(

*z *+ 1 ) =

*zp *+ 1 to get the cyclic group or the dihedral
If

*f *is of type (ii), then Fried showed ([

**8, Section 3**]) that

*f /*
remark that Fried did this without actually knowing the occurring groups (evento prove that there are only finitely many examples seems to require the classi-fication of the finite simple groups).

Now we discuss the type (iii). The group S

*n *is in some sense the generic case.

To get this group take for instance

*f *(

*X*) =

*Xn −X*. The discriminant of

*f *(

*X*)

*−t*is a polynomial of degree

*n−*1 in

*t*, and the roots of the discriminant are preciselythe finite branch points of

*f *: P1

*→ *P1. In this case the discriminant has

*n − *1different simple roots, therefore

*σi *is a transposition for

*i *= 1

*, . . . , r − *1, see 2.1.

Thus

*G *is a transitive group generated by transpositions, and such a group issymmetric.

Similarly we get the alternating group. Choose

*f *such that its derivative
equals (

*Xm − *1)2, thereby 2

*m *+ 1 =

*n*. Denote by

*S *the set of

*m*th roots ofunity. Then the discriminant of

*f *(

*X*)

*− t *equals, up to a multiplicative constant,∆(

*t*) =
(

*t − f *(

*ζ*))2. Note that ∆ has

*m *different roots, each of multiplicity
2. Now

*f *(

*ζ*) =

*f *(

*ζ*) = 0 =

*f *(

*ζ*) for each

*ζ ∈ S*. This shows that (in the

notation of 2.1)

*r *=

*m *+ 1 and each

*σi *(

*i *= 1

*, . . . , r − *1) is a 3-cycle. So the

transitive group

*G *is generated by 3-cycles, thus

*G *= A

*n *(see [

**18, lemme 1 in**

4.]).

Next we exclude

*M*11,

*M*23 and PGL2(7) using the Lemma from above. Ob-
serve that every automorphism of these groups is inner, hence ˆ
PRIMITIVE MONODROMY GROUPS OF POLYNOMIALS
cases. To exclude the two Mathieu groups it suffices to see that no element of

order 11 (resp. 23) is conjugate to its inverse. This can be deduced from [

**2**].

Suppose that PGL2(7) meets the conclusion of the Theorem. As

*Z *has 4
generators, 8

*· *4 =

*|*N

*G*(

*Z*)

*| *does divide 7

*· *48 =

*|G|*, a contradiction.

For the group PΓL2(8) we got two different types of branch cycle descriptions.

We show that the case with

*|σ*1

*| *= 2,

*|σ*2

*| *= 3 does not occur. Of course

*σ*1

*∈ *PGL2(8) and

*σ*2

*/*
*∈ *PGL2(8). Thus the 9–cycle

*σ *=

*σ*1

*σ*2 is not contained
in PGL2(8). Thus

*σ *has order 3 modulo PGL2(8), and therefore cannot beconjugate to its inverse.

It remains to show that PGL2(5), PΓL2(8) of type (3

*, *3 : 9), and PΓL2(9) are
monodromy groups of polynomials with rational coefficients and to exhibit thecorresponding polynomials.

**3.2 The group PGL**2(5)

**. **We know from our result in section 2, that there

is a polynomial

*f ∈ *C[

*X*] with monodromy group PGL2(5). We just compute it,and it will turn out that it can be chosen with rational coefficients. Recall thedefinition of the generators

*σ*1

*, . . . , σs *of the monodromy group. In our case wehave (up to simultaneous conjugation with elements in S6 and reordering the

*σ*’s)

*σ*1 = (1

*, *2)(3

*, *4) and

*σ*2 = (3

*, *2

*, *5

*, *6); that is a consequence of the considerationsin 2.6.

Let

*f *be monic, and let 0 be the branch point corresponding to

*σ*2. Without
loss, above 0 lies the 4-fold point 0, and the simple points

*κ*1 and

*κ*2 (

*κ*1

*, κ*2 = 0).

We have

*κ*1 +

*κ*2 = 0, for otherwise

*f *were a composition with a quadraticpolynomial. We may assume

*κ*1 +

*κ*2 =

*−*6. Then

*f *(

*X*) =

*X*4(

*X*2 + 6

*X *+

*p*)with

*p ∈ *C. The finite branch points of

*f *are the zeroes of

*f *. We have

*f *(

*X*) =2

*X*3(3

*X*2 + 15

*X *+ 2

*p*). Let

*λ*1 and

*λ*2 be the zeroes of

*h*(

*X*) = 3

*X*2 + 15

*X *+ 2

*p*.

They are different, and have the same images under

*f *. Write

*f *=

*q · h *+

*r*with polynomials

*q *and

*r*, such that deg

*r ≤ *1. Then

*f *(

*λi*) =

*r*(

*λi*), hence

*r*(

*λ*1) =

*r*(

*λ*2). Thus

*r *is a constant. On the other hand, by dividing thepolynomials, we get that the coefficient of

*X *in

*r *is 8

*/*3(

*p − *75

*/*8)(

*p − *25). Thechoice

*p *= 75

*/*8 yields 3125

*/*128 as the second finite branch point. However,

*f *(

*X*)

*− *3125

*/*128 = 1

*/*128(16

*X*3

*− *24

*X*2 + 30

*X − *25)(2

*X *+ 5)3 shows that theramification above this point is the wrong one. Thus

*p *= 25.

**3.3 The groups P**Γ

**L**2(8)

**and P**Γ

**L**2(9)

**. **The polynomials have been com-

puted by Matzat. See [

**17, 8.5**] for PΓL2(8) and [

**17, 8.7**] for PΓL2(9).

**Appendix: The Theorems of Ritt**
The setup from section 2.1 allows for short proofs of the classical Theorems of
Ritt about decompositions of polynomials. Throughout this section we deal with

polynomials with complex coefficients. Via model theory the assertions hold for

any algebraically closed field of characteristic 0. Using [

**12, 3.5**] one readily gets

that R.1 and R.2 hold for arbitrary fields of characteristic 0. By a maximal

decomposition of a polynomial

*f *we mean a decomposition

*f *=

*f*1

*◦ f*2

*◦ · · · ◦ fr*where the

*fi *are non–linear and indecomposable.

Theorem R.1 (Ritt).

*Let f *=

*f*1

*◦ · · · ◦ fr *=

*g*1

*◦ · · · ◦ gs be two maximal*
*decompositions of a polynomial f ∈ *C[

*X*]

*. Then r *=

*s and the degrees of thefi’s are a permutation of the degrees of the gi’s. Furthermore, one can pass fromone decomposition to the other one by altering two adjacent polynomials in eachstep.*
From the latter part of this Theorem the question arises when

*a◦b *=

*c◦d *with
indecomposable polynomials

*a*,

*b*,

*c*, and

*d*. Recall that the Cebychev polynomial

*Tn *is defined by

*Tn*(

*Z *+ 1

*/Z*) =

*Zn *+ 1

*/Zn*.

Theorem R.2 (Ritt).

*Let a, b, c, and d be non–linear indecomposable poly-*
*nomials such that a ◦ b *=

*c ◦ d. Assume without loss *deg

*a ≥ *deg

*c. Then thereexist linear polynomials L*1

*, L*2

*, L*3

*, and L*4

*such that one of the following holds.*
*a *=

*c ◦ L*1

*, b *=

*L−*1

*◦ d, (the uninteresting case).*
*L*1

*◦ a ◦ L−*1 =

*Xk · t*(

*X*)

*m, L*
*L*1

*◦ c ◦ L−*1 =

*Xm, L*
4

*◦ d ◦ L*3 =

*X k · t*(

*X m*)

*L*1

*◦ a ◦ L−*1 =

*T*
*L*2

*◦ b ◦ L*3 =

*Tn*
*L*1

*◦ c ◦ L−*1 =

*T*
*L*4

*◦ d ◦ L*3 =

*Tm .*
*for Cebychev polynomials Tm and Tn.*
Let

*f *be a polynomial with complex coefficients, and let

*x *be a transcendental
over C. Set

*t *=

*f *(

*x*), and let Ω be the Galois closure of C(

*x*)

*|*C(

*t*). Let

*G *=Gal(Ω

*|*C(

*t*)) be the monodromy group of

*f *, and let

*U *be stabilizer of

*x*. We viewtwo decomposition of

*f *as equivalent, if they differ just by linear twists (like (1)in R.2). As an easy consequence of L¨
uroth’s theorem, we see that the equivalence
classes of maximal decompositions of

*f *correspond bijectively to the maximalchains of subgroups from

*U *to

*G*. If

*f *=

*f*1

*◦ · · · ◦ fr *is such a decomposition,then the associated chain of subgroups is

*U *=

*U*0

*< U*1

*< . . . < Ur−*1

*< Ur *=

*G*,where

*Ui *is the stabilizer of

*f*1(

*f*2(

*· · · *(

*fi*(

*x*)

*· · · *)). Then Theorem R.1 is a directconsequence of
Theorem R.3.

*Let G be a finite group with subgroups U and C such that G *=

*U C and C is abelian. Then the maximal chains of subgroups from U to G haveequal lengths and (up to permutation) the same relative indices. Furthermore,one can pass from one chain to an other one just by changing one group in thechain in each step.*
**Proof of R.3**. Choose a minimal counter–example subject to

*U *+ [

*G *:

*U *]

being minimal. Let

*U *=

*A*0

*< A*1

*< . . . < Ar *=

*G *and

*U *=

*B*0

*< B*1

*< . . . <*
PRIMITIVE MONODROMY GROUPS OF POLYNOMIALS

*Bs *=

*G *be two chains failing the assertion. Then

*A*1 =

*B*1 and core

*G*(

*U *) = 1,where core

*G*(

*U *) means the maximal normal subgroup of

*G*, which is contained in

*U *. Set

*NA *= core

*G*(

*A*1) and

*NB *= core

*G*(

*B*1). These groups are non–trivial, as

*G *=

*A*1

*C*, hence 1 =

*A*1

*∩ C ≤*
in

*A*1, and

*NA ≤ U *, we get

*A*1 =

*U NA*. Likewise

*B*1 =

*U NB*. Set

*D *=

*A*1

*, B*1 .

The assumptions of the Theorem are fulfilled if we replace

*U *by

*A*1 or

*B*1. Weconsider, in addition to the given chains, a maximal chain from

*D *to

*G*. Thus weare done once we know that

*A*1 and

*B*1 are maximal in

*D*, [

*D *:

*A*1] = [

*B*1 :

*U *],and [

*D *:

*B*1] = [

*A*1 :

*U *]. Note that

*D *=

*NAU, B*1 =

*NAB*1. Suppose there is agroup

*X *properly between

*A*1 and

*D *=

*NAB*1. Then

*X *=

*NA*(

*X ∩ B*1), hence

*U < X ∩ B*1

*< B*1, a contradiction. By symmetry

*B*1 is maximal in

*D *as well.

Finally we have

*NA ∩B*1

*≤ A*1

*∩B*1 =

*U *, hence

*NA ∩B*1 =

*NA ∩U *. This yields
[

*D *:

*B*1] = [

*NA *:

*NA ∩ B*1] = [

*NA *:

*NA ∩ U *] = [

*A*1 :

*U *]. Again by symmetry weget [

*D *:

*A*1] = [

*B*1 :

*U *].

**Proof of R.2**. As in 2.1, let

*G *be the Galois group of the Galois closure of

C(

*X*)

*|*C(

*t*), where

*a*(

*b*(

*X*)) =

*c*(

*d*(

*X*)) =

*t*.

Let

*A*,

*B*, and

*U *be the fix groups of

*b*(

*X*),

*d*(

*X*), and

*X*, respectively. The
case

*A *=

*B *yields (1) of the Theorem. From now on assume

*A *=

*B*. Then

*U *=

*A ∩ B *is core–free in

*G*, and the chains of subgroups

*U ⊂ A ⊂ G *and

*U ⊂ B ⊂ G *are maximal. Let

*Z *be a cyclic complement of

*U *in

*G *(c.f. 2.1).

Set

*NA *= core

*G*(

*A*) (= 1, see the proof of R.3),

*NB *= core

*G*(

*B*).

Then, by the maximality of the chains above,

*G *=

*ANB *=

*BNA*,

*A *=

*U NA*,
Set

*m *= [

*G *:

*A*] = [

*B *:

*U *],

*n *= [

*G *:

*B*] = [

*A *:

*U *].

Claim 1.

*The monodromy groups of b and c are the same, as well as the ones*
Proof. Set

*N *= core

*B*(

*U *). Of course

*B ∩ NA *=

*U ∩ NA ≤ N *. On the
other hand, the set of

*G*-conjugates of

*A *is the same as the set of

*B*-conjugatesof

*A*. Therefore

*N ≤ NA*, hence

*N ≤ B ∩ NA*. This shows

*B ∩ NA *=

*N *. Now

*G/N*
*A *=

*BNA/NA *=

*B/B ∩ NA *=

*B/N *yields the assertion.

Now we are going to study three different permutation representations of

*G*.

First let

*G *act on the cosets of

*U *. Then the set of cosets of

*A *provides a systemof imprimitivity, and so does the set of cosets of

*B*. The intersection of a cosetof

*A *and a coset of

*B *is a coset of

*U *: Without loss consider

*A *and

*Bg*. We mayassume

*g ∈ NA ⊆ A *(as

*G *=

*BNA*). Then

*U g ≤ A ∩ Bg*. If

*U h ≤ A ∩ Bg*, then

*hg−*1

*∈ A ∩ B *=

*U *, hence

*U h *=

*U g*. Therefore

*U g *=

*A ∩ Bg*.

Denote by

*πA *the canonical homomorphism

*G −→ G/NA ≤ *Sym(

*G/A*) of
permutation groups, likewise for

*B*. Note that

*πA*(

*G*) and

*πB*(

*G*) act primitivelyby the maximality of

*A *and

*B *in

*G*.

Let ind(

*g*),

*o*(

*g*), and f(

*g*) be the index of

*g*, the number of cycles of

*g*, and
the number of fixed–points of

*g*. Define ind

*A*(

*g*),

*o*(

*g*), and f

*A*(

*g*) analogously for

*πA*(

*g*), likewise for

*B*. From the considerations above we get f(

*g*) = f

*A*(

*g*)

*· *f

*B*(

*g*).

For a fixed

*g ∈ G *let [

*ν*1

*, ν*2

*, · · · , νk*] be the cycle type of

*πA*(

*g*) (i.e.

*πA*(

*g*) hascycles of lengths

*ν*1,

*ν*2,

*. . . *) and [

*µ*1

*, µ*2

*, · · · , µl*] be the cycle type of

*πB*(

*g*). Then

*k *=

*oA*(

*g*) =

*m − *ind

*A*(

*g*)

*l *=

*oB*(

*g*) =

*n − *ind

*B*(

*g*)
(

*νi, µj*) =

*o*(

*g*) =

*mn − *ind(

*g*)

*.*
ind(

*g*)

*≥ n · *ind

*A*(

*g*)
ind(

*g*)

*≥ m · *ind

*B*(

*g*)

*.*
Let

*g*1,

*g*2,

*. . . *,

*gs *be a generating system of

*G *according to 2.2(*).

*A*(

*gu*) =

*m − *1

*,*
*B *(

*gu*) =

*n − *1

*.*
ind

*A*(

*gu*)

*≥ m − *1 (for otherwise the elements

*πA*(

*gu*) were a
branch cycle description of a cover

*X → *P1 with

*X *having negative genus).

On the other hand,
(

*mn − *1) =

*m − *1 . This
proves the assertion. Here and in the following we use implicitly the symmetryof certain assertions in

*A *and

*B*.

Claim 3.

*If πA*(

*G*)

*is not cyclic, then *ind(

*g*)

*≥ m · *ind

*B*(

*g*) + ind

*A*(

*g*)

*for all*
*g ∈ {g*1

*, g*2

*, . . . , gs}.*
Proof. Assume the contrary, which implies

*o*(

*g*)

*> m · oB*(

*g*) +

*oA*(

*g*)

*− m *for
some

*g ∈ {g*1

*, g*2

*, . . . , gs}*. Assign to this

*g *the

*ν*’s and

*µ*’s as above. Then
Thus there is an index

*i*, without loss

*i *= 1, such that
Let

*T *be the number of

*j*’s such that

*ν*1 does not divide

*µj*. Then
(

*ν*1

*− *(

*ν*1

*, µj*))

*< ν*1

*− *1

*,*
hence

*T ≤ *1. Thus there is at most one

*j*0 such that

*ν*1 does not divide

*µj *. But
(

*ν*1

*, µj *) = 1 yields the contradiction 0

*< *0. Therefore the

*µ*’s have a common
divisor

*δ > *1. From Claim 2 we know that the elements

*πA*(

*g*1)

*, . . . , πA*(

*gs*)provide a branch cycle description of a polynomial. A common divisor

*δ *of the

*µ*’s means, that this polynomial has the form

*h*(

*X*)

*δ *+

*e *for some polynomial

*h*
PRIMITIVE MONODROMY GROUPS OF POLYNOMIALS
and a constant

*e*. However, this polynomial is decomposable, contrary to

*πA*(

*G*)being primitive.

Claim 4.

*If πA*(

*G*)

*is not cyclic, then *ind(

*g*) =

*m · *ind

*B*(

*g*) + ind

*A*(

*g*)

*for all*
*g ∈ {g*1

*, g*2

*, . . . , gs}.*
Proof. Suppose wrong. Then Claim 2 and Claim 3 yield the contradiction
ind

*A*(

*gu*) =

*m*(

*n−*1)+

*m−*1 =

*mn−*1

*.*
Claim 5.

*Exactly one of the following holds.*
(1)

*πA*(

*G*)

*or πB*(

*G*)

*is cyclic.*

(2)

*G, πA*(

*G*)

*, and πB*(

*G*)

*are dihedral and act naturally (i.e. the cyclic group*
*of index *2

*acts regularly in each case).*
Proof. Suppose that (1) doesn’t hold.

Choose any

*g ∈ {g*1

*, g*2

*, . . . , gs}*.

Assign to

*g *the cycle lengths

*νi *and

*µj *of

*πA*(

*g*) and

*πB*(

*g*) as in the proof ofClaim 3.

*νi *+ 1

*− νj *for each

*i *= 1

*, . . . , k .*
Furthermore, also by this proof, the following holds: For each

*j *there is at mostone

*i *such that

*µj *does not divide

*νi*. In particular, for fixed

*j*,

*νi ≥ µj *besidesat most one index

*i*. Thus
(

*oA*(

*g*)

*− *1)

*µj *+ 1

*≤ m .*
Now, for

*gu *in

*{g*1

*, g*2

*, . . . , gs}*, let

*wu *be the maximal associated cycle length

*µj*. Then
(

*m − *ind

*A*(

*gu*)

*− *1)

*wu *+ 1

*≤ m .*
Dividing by

*wu *and adding for

*u *= 1

*, *2

*, . . . , s *yields
Therefore all besides two

*w*’s are 1, and these two exceptions are 2. By the choiceof the

*w*’s, this implies the existence of two indices

*u*1 and

*u*2 such that

*πB*(

*gu *)
and

*πB*(

*gu *) are involutions, and the other

*π*
*B *(

*gu*)’s are trivial. The same holds
for the images of

*gu *in

*πA*(

*G*) for the same indices

*u*, as can be seen (for instance)by Claim 4. Finally, as

*G −→ πA*(

*G*)

*× πB*(

*G*) is an injective homomorphism,the involutions

*gu *and

*g*
are the only non–trivial elements in

*{g*
We get the assertion about the action as follows:

*gu g*
(neither of which is contained in

*g*
a dihedral group of order 2

*mn*.

The final step is to formulate our results in terms of polynomials: Suppose
that (1) in Claim 5 holds. Then, by Claim 1, we need to study the decomposition

*p*(

*Xm*) =

*q*(

*X*)

*m *for some polynomials

*p *and

*q*. Set

*p*(

*X*) =

*Xk · r*(

*X*) with

*r *apolynomial such that

*r*(0) = 0. Then

*Xkm · r*(

*Xm*) =

*q*(

*X*)

*m*. Thus

*Xk *divides

*q*(

*X*), hence

*q*(

*X*) =

*Xk · s*(

*X*) with a polynomial

*s *such that

*s*(0) = 0. We get

*r*(

*Xm*) =

*s*(

*X*)

*m*. Now every zero of

*r *occurs with a multiplicity divisible by

*m*,hence

*r*(

*X*) =

*t*(

*X*)

*m *with a polynomial

*t*. But then

*s*(

*X*) =

*ζt*(

*Xm*) for some

*m*-th root

*ζ *of 1. Substituting back we get our result.

Now assume that case (2) of Claim 5 holds. Without loss of generality we
assume

*a◦b *=

*c◦d *=

*Tmn *=

*Tm◦Tn*. Then the fix groups of

*Tn*(

*X*) and of

*b*(

*X*) in

*G *have the same order, thus they are equal (every group

*M *between

*U *and

*G *isuniquely determined by its order, as

*M *=

*U Z ∩M *=

*U *(

*M ∩Z*) and subgroups ofcyclic groups are determined by their order). Thus C(

*Tn*(

*X*)) = C(

*b*(

*X*)), hence

*b *=

*L−*1

*◦ T*
*n *for some linear polynomial

*L*2. Then

*Tn ◦ Tm *=

*a ◦ b *=

*a ◦ L−*1
Analogously express

*c *and

*d *in terms of

*Tm *and

*Tn*. The assertion follows.

M. Aschbacher,

*On conjectures of Guralnick and Thompson*, J. Algebra

**135 **(1990),

277–343.

J. Conway, R. Curtis, S. Norton, R. Parker, R. Wilson,

*Atlas of Finite Groups:Maximal Subgroups and Ordinary Characters for Simple Groups*, Clarendon Press,Oxford, New York, 1985.

F. Dorey, G. Whaples,

*Prime and composite polynomials*, J. Algebra

**28 **(1974), 88–

101.

W. Feit,

*On symmetric balanced incomplete block designs with doubly transitive au-*

tomorphism groups, J. of Comb. Theory Series A

**14 **(1973), 221–247.

W. Feit,

*Some consequences of the classification of finite simple groups*, The SantaCruz conference on finite groups, Proc. Sympos. Pure Math., vol. 37, AMS, Provi-dence, Rhode Island, 1980, pp. 175–181.

W. Feit,

*E-mail from 28. Jan 1992*.

M. Fried,

*On a conjecture of Schur*, Michigan Math. J

**17 **(1970), 41–55.

,

*The field of definition of function fields and a problem in the reducibility of*
*polynomials in two variables*, Illinois Journal of Mathematics

**17 **(1973), 128–146.

,

*Rigidity and applications of the classification of simple groups to mon-*
*odromy, Part II – Applications of connectivity; Davenport and Hilbert-Siegel Prob-lems*, Preprint.

,

*Review of Serre’s ‘Topics in Galois Theory’*, Bull. Amer. Math. Soc.

**30(1)**
M. Fried, M. Jarden,

*Field Arithmetic*, Springer, Berlin Heidelberg, 1986.

M. Fried, R. E. MacRae,

*On the invariance of chains of fields*, Illinois Journal of

Mathematics

**13 **(1969), 165–171.

D. Gorenstein,

*Finite Groups*, Harper and Row, New York–Evanston–London., 1968.

R.M. Guralnick, J. Saxl,

*Monodromy groups of polynomials*, Preprint (1993).

R.M. Guralnick, J.G. Thompson,

*Finite groups of genus zero*, J. Algebra

**131 **(1990),

303–341.

*osbare Gruppen*, Arch. Math.

**6 **(1955), 303–310.

B. H. Matzat,

*Konstruktion von Zahl– und Funktionenk¨*
*loisgruppe*, J. Reine Angew. Math.

**349 **(1984), 179–220.

*eres de *Q(

*T *)

*de groupe de Galois *˜

*An*, J. Alg.

**131**
PRIMITIVE MONODROMY GROUPS OF POLYNOMIALS
uller,

*Monodromiegruppen rationaler Funktionen und Irreduzibilit¨*
*nomen mit variablen Koeffizienten*, Thesis (1994).

olklein,

*On a problem of Davenport*, submitted (1994).

J. F. Ritt,

*Prime and composite polynomials*, Trans. Amer. Math. Soc.

**23 **(1922),

51–66.

H. Wielandt,

*Finite Permutation Groups*, Academic Press, New York and London,1964.

*E-mail address*: mueller@mi.uni-erlangen.de

Source: http://www.mathematik.uni-wuerzburg.de/~mueller/Papers/mon.pdf

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Considerando a dinâmica que a Câmara Municipal da Moita tem vindo a imprimir no domínio das actividades turístico-culturais, o Posto de Turismo passou a dispor de uma Galeria de exposições, um espaço inovador no centro da vila da Moita onde munícipe e visitante poderão apreciar exposições temporárias de temática variada. Pretendendo-se que a Galeria de Exposições seja também um