sign in sign up
<<
Home , Frobenius group

J. Theory 16 (2013), 707–718 DOI 10.1515/jgt-2013-0015 © de Gruyter 2013

Five wild nearfields

Theo Grundhöfer Communicated by Linus Kramer

Abstract. The projective groups PSL2.FpŒt/ with p 2; 3 and three related groups have normal that are sharply transitive on the2 ¹ correspondingº projective line. This leads to five infinite nearfields which are not Dickson nearfields.

1 Introduction

For p 2; 3 the SL2 Fp is solvable. As a consequence, the 2 ¹ º commutator of the projective group PSL2.FpŒt/ over the polynomial ring FpŒt is sharply transitive on the projective line Fp.t/ . A similar re- [ ¹1º sult holds for three other affine coordinate rings of curves over finite fields; see Theorem 2.2. These sharply transitive actions on projective lines lead in Theorem 3.1 to five infinite nearfields (with characteristic 2 or 3) that are wild in the sense that they are not Dickson nearfields.

2 Frobenius groups on projective lines

A is called semi-regular if every point stabilizer is trivial. A Frobenius group is a transitive permutation group such that the stabilizer of any two distinct points is trivial (or equivalently, such that some point stabilizer is semi-regular on the other points). Let R be a principal ideal domain with field of fractions K. Then the special linear group SL2 R acts transitively on the projective line PG1K K , by WD [ ¹1º Bézout’s lemma. If R has at most two units, then id are the only diagonal ma- ˙ trices in SL2 R, hence the stabilizer PSL2.R/ of is faithfully induced by the 1 R  1 1 group 0 1 and therefore semi-regular on the other points. Thus PSL2 R acting on PG1K is a Frobenius group. Theorem 2.2 shows that in some cases this infinite Frobenius group has a Frobenius , i.e. a sharply transitive . The following lemma says that one should expect such a Frobenius kernel only in characteristic 2 or 3. 708 T. Grundhöfer

Lemma 2.1. Let K be a field with a subring R 0 . Assume that the group ¤ ¹ º PSL2 R normalizes a subgroup H of PGL2 K that is sharply transitive on the projective line PG1K. Then K has characteristic 2 or 3, and R contains no subfield of cardinality at least 4.

Proof. Writing G PSL2 R we have WD HG H Ì .HG/ : D 1 The stabilizer .HG/ .PGL2 K/ is induced by upper triangular matrices 1 Ä 1 and therefore metabelian, hence HG=H .HG/ is metabelian. We infer that Š 1 G H. Hence all unipotent elements of G are trivial. 00 Ä 00 If K has characteristic p > 3, then R has a subfield F with F 4; thus j j  G .PSL2 F/ PSL2 F , which contains nontrivial unipotent elements. If K 00  00 D has characteristic 0, then G00 .PSL2 Z/00, and .SL2 Z/00 contains the unipotent 1 6   matrix 0 1 , see [17, Lemma 2]. We write Gn gn g G if G is a group and n is an . WD h j 2 i

Theorem 2.2. (a) Let G PSL2.FpŒt/ with p 2; 3 . Then the commutator WD 2 ¹ º group G0 is the unique normal subgroup of G that is sharply transitive on the p projective line PG1K over K Fp.t/, and G G . D 0 D (b) Let G PSL2 R where R is one of the two rings WD 2 3 2 3 R2 F2Œs; t=.s s t t 1/ and R3 F3Œs; t=.s t t 1/ WD C C C C WD C C with indeterminates s; t over the prime field Fp of R. Let K be the field of fractions of R. Then Gp is the unique normal subgroup of G that is sharply p transitive on the projective line PG1K, and G0 < G .

(c) Let G PSL2 R5 where WD 2 5 3 R5 F2Œs; t=.s s t t 1/ WD C C C C with indeterminates s; t over the prime field F2 of R5, and let K be the field of fractions of R5. Then G has a normal subgroup that is sharply transitive on 2 2 the projective line PG1K and contains G (and G G ). 0 Ä Proof. The polynomial rings R FpŒt in (a) and the rings R in (b), (c) are prin- D cipal ideal domains; see [9, Theorem 5.2]. We infer that G PSL2 R is tran- D sitive on PG1K. The polynomials that define R2, R3, R5 are absolutely irre- ducible, hence Fp is the full constant field of each of the five rings R (compare [9, Corollary 2.4 (i)]). Thus the group of units of R is R F 1 , see  D p D ¹˙ º [12, Lemma 1.1]. Hence the stabilizer G is induced by unipotent upper trian- 1 Five wild nearfields 709 gular matrices and is therefore an elementary abelian p-group. The sharply tran- sitive subgroups H G are characterized by the two conditions HG G and Ä 1 D H G 1 . \ 1 D ¹ º 0 1  1 1  Let ˛ G and  G be induced by the matrices 1 0 and 0 1 , respec- 2 2 2 1 3 p tively. Then ˛ 1 .˛/  are defining relations for ˛;  PSL2 Fp, D D D h i D which is isomorphic to the S3 for p 2 and isomorphic to the D A4 for p 3. Moreover D ´ ˛ for p 2; ˛;  0 h i D h i D ˛; ˛ for p 3: h i D Thus PSL2 Fp ˛;  ˛;  Ì  . D h i D h i0 h i Let B B.R/ denote the group of all upper triangular matrices in SL2.R/ and D 1 1  put B0 B SL2 Fp id; . WD \ D h 0 1 i (a) A theorem of Nagao [15] yields the decomposition

SL2.FpŒt/ SL2 Fp B B D  0 as an amalgamated product; see also [20, Section II.1.6, Exercise 2, p. 88], or [18, p. 19] for a quick deduction from a version of the Ping-Pong Lemma. There- fore G PSL2.FpŒt/ ˛;   G : D D h i h i 1 The subgroup G0G contains ˛;  0  ˛;  and G , hence G0G G. 1 h i h i D h i 1 1 D From the description of ˛;  PSL2 Fp given above we obtain a group homo- 2 h i D 3 p morphism ' ' ˛;   G with './  and '.˛/  (which D W h i ! h i Ä 1 D D is  for p 2 and 1 for p 3). By the universal property of the amalgamated D D product G ˛;   G we can extend ' and the identity  G G to D h i h i 1 W 1 ! 1 a group epimorphism  '  G G . Since G is abelian, the group G0 D  W ! 1 1 is contained in the kernel of  2 and has therefore trivial intersection with D .G/ G . (Compare [10, Theorem 2.3 (ii)] and [11, Lemma 3.2 (ii)].) D 1 Thus G0 is sharply transitive on PG1K. If H is any sharply transitive normal subgroup of G, then G=H G is abelian, hence G0 H, and then G0 H. Š 1 Ä D The equation G Gp follows from [11, Lemma 3.2 (ii)]. 0 D (b) Our rings R are affine coordinate rings of curves, and the structure of GL2 R and of SL2 R is described by a general theorem of Serre [20, Section II.2.5, The- orem 10, p. 119]; see also [12]. (The transition from GL2 R to SL2 R is straight- forward, see [20, Section I.1.3, Exercise 2, p. 8] or [2, Section 8.5, Theorem 27, p. 211].) For R Rp R2;R3 the corresponding curve is elliptic (without Fp-rational D 2 ¹ º points except one point at infinity). We have the decomposition  SL2 R SL2 Fp B B  D  0  710 T. Grundhöfer

where  SL2 R is the free product of p cyclic groups of p 1; see Ä C [13, Theorem 5.3], which relies on results in [21], or [20, Section 2.4.4, p. 115 and Exercise 3, p. 117 and Exercise 3, p. 120]. The free factor  cannot contain the central involution id, hence identifying  with its in G we obtain  G PSL2 R ˛;   G : D D h i h i 1  The groups  and ˛;  ˛;  p are contained in Gp. From  ˛;  ˛;  h i0 D h i h ih i0 D h i we infer that the subgroup G Gp coincides with G. The quotient 1 G=Gp G Gp=Gp G =.G Gp/ D 1 Š 1 1 \ is abelian, hence G Gp. In fact, G < Gp since the free factor  is not con- 0 Ä 0 tained in G0 (consider G=G0 and use [20, Section I.1, Proposition 4, p. 6]). As in the proof for part (a) we obtain a group epimorphism  2 G G D W ! 1 which is trivial on Gp. Hence G Gp 1 , and Gp is sharply transitive on 1 \ D ¹ º the projective line. If H is any sharply transitive normal subgroup of G, then G=H G is an elementary abelian p-group, hence Gp H, and then Gp H. Š 1 Ä D (c) The curve associated with the ring R5 is hyperelliptic of genus 2. (The fol- lowing arguments establish assertion (c) also for the rings F2Œt, R2 and R5, and with small complications one could include also F3Œt and R3, using [14, Sec- tions 2.3, 2.13].) Note that G B. 1 D The group G PSL2 R5 GL2 R5 acts on the associated Bruhat–Tits tree X D D (which here is the regular tree with valency 3), see Serre [20, Section I.1.3]. The principal ideal domain R5 has trivial Picard group (ideal class group), hence the quotient graph G X has only one cusp in the sense of Serre [20, pp. 104–106]. n By [20, Theorem 10, p. 119] the group G is an amalgamated product

G ƒ ƒ B B D  \ where ƒ 1Y is the fundamental group of Y , and Y is a graph of groups (in D fact, Y is a finite graph of finite groups; see [20, Section II.2.3, Corollary 4, p. 108] and [20, Section II.1.3, Proposition 4, p. 76] combined with [1, Section XII.2, Theorem 2, p. 230]). Loosely speaking, this means that ƒ is an amalgamated product of vertex stabilizers Gv G with amalgamation along the stabilizers Ä G G G of edges e .v; v / of Y . e v v0 0 D \ D 2 The elements of any vertex stabilizer Gv have characteristic polynomial x 1 2 C or x x 1, see [14, Lemma 1.3]. If Gv is not an elementary abelian 2-group, C C then some of its elements have characteristic polynomial x2 x 1, and then C C Gv 3 or Gv GL2 F2 S3 by [14, Sections 2.6, 2.12]. j j D Š Š We claim that for every vertex or edge x of Y , there exists a unique epimor- 2 phism 'x ' Gx Gx B. This is obvious if Gx B is trivial, in par- D x W ! \ \ ticular if Gx 3, and also if Gx S3. If Gx is abelian and Gx B 1 , j j D Š \ ¤ ¹ º Five wild nearfields 711

then Gx CG.Gx B/ B, hence 'x idG is the only epimorphism as re- Ä \ D D x quired. By the uniqueness, 'e coincides with the restriction 'v Ge for every edge j e of Y containing the vertex v. Thus we obtain a group epimorphism  2 G B, see [3, Lemma 7.2, D W ! p. 31] or [2, Section 8.3, Lemma 20, p. 201]. The kernel H of  is a normal com- plement to B G , i.e. a sharply transitive normal subgroup. Since G=H B is D 1 Š an elementary abelian 2-group, we have H , and G G2 since G=G2 has Ä 0 Ä exponent 2 and is therefore abelian. (In order to decide whether or not H G2 D one would need more information on the graph Y .)

Remarks. (1) In addition to the five rings in Theorem 2.2, there exists just one further principal ideal domain among the affine coordinate rings of curves over fi- 2 3 nite fields, namely R F4Œs; t=.s s t c/ with c F2; see [8]. This D C C C 2 n ring R has 3 units, hence PSL2 R is not a Frobenius group on the corresponding projective line. Moreover, this permutation group PSL2 R has no sharply transitive normal subgroup by Lemma 2.1. For K Q and for the imaginary quadratic number fields K Q.p d/ with D D d 2; 7; 11; 19; 43; 67; 163 , the ring R of all algebraic in K is a princi- 2 ¹ º pal ideal domain with only two units. Thus PSL2 R is a Frobenius group on PG1K, but this Frobenius group has no sharply transitive normal subgroup by Lemma 2.1. Still, these groups PSL2 R might contain non-normal subgroups that are sharply transitive on PG1K; see [16] and [4] for K Q and R Z. D D (2) The group PSL2.F2Œt/ GL2.F2Œt/ happens to be a Coxeter group: in the D presentation given in [20, Section II.1.7, Exercise 4, p. 88], the commutativity of 1 F2Œt  the involutory generators Tn of B 0 1 can be expressed by the relations 2 D .TmTn/ 1. All groups G in Theorem 2.2 are virtually free of finite rank, see D [20, Section II.2.6, Corollary to Proposition 11, p. 121]. For p 2; 3 the second 2 ¹ º commutator group PSL2.FpŒt/00 is free of infinite rank, see [11, p. 294].

3 Nearfields

Nearfields are generalizations of skew fields: just drop one of the two distributive laws. Every nearfield N .N; ; / with the distributive law D C  a .b c/ a b a c  C D  C  is a vector space over the skew field

K k N .a b/ k a k b k for all a; b N ; WD ¹ 2 j C  D  C  2 º which is called the kernel of N . Such a nearfield N .N; ; / is said to be D C  712 T. Grundhöfer a Dickson nearfield, if .N; / can be made into a skew field D .N; ;/ such C D C that each left nearfield multiplication x a x with a 0 belongs to the group 7!  ¤ €L1D x ax a D ; Aut D . Dickson nearfields are “tame” in the WD ¹ 7! j 2  2 º sense that they are closely related to skew fields. According to Zassenhaus [23], every finite nearfield is either a Dickson nearfield or isomorphic to one of seven “wild” examples; see also [19, Section 20.3] or [22]. Constructions of Dickson nearfields are described in [7] and [22]. For infinite non- Dickson nearfields see [4, 5, 24].

Theorem 3.1. Let R be one of the five rings FpŒt, R2, R3, R5 in Theorem 2.2, with characteristic p 2; 3 , and let K be the field of fractions of R. Then there 2 ¹ º exists a nearfield N with the following properties: (1) N has dimension 2 over its kernel K, which coincides with the center of N . (2) The N , acting on N K2, contains the linear group  D K .SL2 R/ GL2 K; for the two polynomial rings R FpŒt one has in  0 Ä D fact equality N Fp.t/ SL2.FpŒt/ .  D  0 (3) N is wild, i.e. not a Dickson nearfield. These five nearfields are mutually not isomorphic.

Proof. According to Theorem 2.2 there exists a group H PGL2 K that is sharply Ä transitive on PG1K and contains .PSL2 R/0. The full preimage H GL2 K of H 2 Ä is sharply transitive on K 0 , and K .SL2 R/ H. (The group K .SL2 R/ n ¹ º  0 Ä  0 is a direct product for p 2, and for p 3 it is the direct product of .SL2 R/ D D 0 with any subgroup of index 2 in K which does not contain 1.) For the two  polynomial rings R FpŒt we have the equality H K .SL2 R/ . D D  0 There exists a nearfield N .K2; ; / with multiplicative group N H as D C   Š in assertion (2): choose a nonzero vector 1 K2 and define “ ” by h.1/ x h.x/ 2   D and 0 x 0 for h H, x K2.  D 2 2 For assertion (1), we observe that K is contained in the center of N , and this center is contained in the kernel of N . If the center or the kernel were larger than K, then N would coincide with its kernel, as dimK N 2. Then N would D be a skew field of dimension 2 over the central subfield K, hence the group N  were commutative. However, R contains a copy of the polynomial ring FpŒt, hence the group N .SL2 R/ contains a copy of the group SL2.FpŒt/ which   0 0 is not commutative (in the notation of the proof for Proposition 3.2, the matrices L1;Lt SL2.FpŒt/ do not commute). This is a contradiction. 2 0 The main task is to prove assertion (3); this is a consequence of Proposition 3.2 below, as R is infinite. Five wild nearfields 713

The five nearfields are mutually not isomorphic, because the five field exten- sions K Fp are mutually not isomorphic: they differ by their characteristic or by j their genus, see [9, Theorem 4.4 or Theorem 5.1].

Proposition 3.2. Let K be field and let R be a subring of K with R 4. Let N be a nearfield with the following properties: j j  (1) N has dimension 2 over its kernel K, which coincides with the center of N . (2) The multiplicative group N , acting on N K2, contains the linear group  D .SL2 R/ GL2 K. 0 Ä Then N is wild, i.e. not a Dickson nearfield. Assumption (2) implies that K has characteristic 2 or 3: the commutator groups .SL2 Fp/ SL2 Fp with p > 3 and .SL2 Z/ contain non-trivial unipotent ele- 0 D 0 ments (see [17, Lemma 2]), hence none of these groups can be contained in N . The assumption R 4 in the proposition is necessary, because the quaternion j j  group .SL2 F3/0 < GL2 F3 is the multiplicative group of a Dickson nearfield of order 9, and .SL2 F2/0 is the multiplicative group of the field F4. Proof. We assume that N .N; ; / is a Dickson nearfield and aim for a contra- D C  diction. Then there exists a skew field D .N; ;/ and a map ˛ N Aut D D C W  ! such that a x ax˛.a/ for a; x N . This map ˛ N Aut D is a group  D 2  W  ! homomorphism by the associative law in N . Steps 1 and 2 below can essentially be found in Zassenhaus [24, pp. 344f. and 356]; we repeat the proofs with some modifications, thus correcting the ar- gument in [5, p. 222, Step 2].

Step 1. For every a K 0 the automorphism ˛.a/ is the identity idD or the 2 n ¹ º conjugation by a. Let a 1 and x N . Then a 1 K 0 , hence ¤ 2 2 n ¹ º .a 1/x˛.a 1/ .a 1/ x x .a 1/ x a x a x x ax˛.a/ x: ( ) D  D  D  D  D 

In this dependence relation for the automorphisms ˛.a 1/, ˛.a/ and idD, we replace x by yx with y N and obtain 2 .a 1/y˛.a 1/x˛.a 1/ ay˛.a/x˛.a/ yx; D and left multiplication of equation . / by y gives  y.a 1/x˛.a 1/ yax˛.a/ yx: D 714 T. Grundhöfer

Subtracting the last two equations leads to

Ax˛.a 1/ Bx˛.a/ D where A .a 1/y˛.a 1/ ya y and B ay˛.a/ ya do not involve x. WD C WD Putting x 1 we obtain A B. If ˛.a 1/ ˛.a/, then equation ( ) shows that D D ˛.a D1/ ˛.a/  ˛.a/ idD. If ˛.a 1/ ˛.a/, then x x for some x N , hence D ¤ ¤ 2 A B 0 and therefore y˛.a/ a 1ya for every y N . D D D 2

Step 2. We may assume that K is the set of all elements fixed by ˛.N /; in partic- ular, K is a subfield of D. Let a; b K. Then a b ab˛.a/ K, hence ab K or ba K by Step 1. 2  D 2 2 2 Thus a a aa K and .a b/2 a2 b2 ab ba K, whence both ab  D 2 C D C 2 and ba belong to K. Moreover, the nearfield inverse of a K 0 is 2 n ¹ º 1 a ˛.a/ a 1; D again by Step 1. Thus K K 0 is a subgroup of the multiplicative group D  WD n ¹ º  of D (and K is a sub-skew-field of D). For a K with ˛.a/ idD we have 2  ¤ xa˛.x/ x a a x ax˛.a/ aa 1xa xa D  D  D D D ˛.x/ by Step 1, hence a a for every x N . Thus every element of the set 1 D 2 K ˛ .idD/ is fixed by ˛.N /. n 1 1 If K ˛ .idD/ , then this set K ˛ .idD/ generates the group K n ¤ ; n 1 with respect to the skew field multiplication (since ˛ .idD/ and K are sub- groups of D), hence K is fixed element-wise by ˛.N /. 1 If K ˛ .idD/, then for a K and x N we have  Â 2  2  ax ax˛.a/ a x x a xa˛.x/: D D  D  D Now we consider the opposite skew field Dop .N; ; / of D with the multi- WD C  plication x y yx, and we define ˇ N Aut Dop Aut D by  WD W  ! D yˇ.x/ xy˛.x/x 1: D Then x y xy˛.x/x 1x x yˇ.x/ for x; y N , and aˇ.x/ a for a K,  D D  2  D 2 x N . Thus replacing .D; ˛/ by .Dop; ˇ/ we achieve that every element of K 2  is fixed by ˛.N /. Clearly all elements of D fixed by ˛.N / belong to the kernel of N , which is K by assumption. This implies that K is a subfield of D. Five wild nearfields 715

Step 3. The group ˛.N / is abelian.

We have ˛.N / AutK D by Step 2, hence it suffices to show that AutK D is  Ä abelian. Observe that D is a (right) vector space over K of dimension 2. This implies that dimC D 4, where C is the center of D; see Jacobson [6, Sec- Ä tion VII.9.1, p. 175] or [5, p. 222]. If D is commutative, then AutK D contains at most two elements and is therefore abelian. If D is not commutative, then K is a maximal subfield of D, hence C K. The  Skolem–Noether Theorem implies that AutK D consists of inner automorphisms of D that fix every element of K. The maximal subfield K of D coincides with its centralizer in D. Hence AutK D is the group of all inner automorphisms induced by elements of K, and therefore isomorphic to the K=C .

Step 4. We obtain a contradiction.

2 Let  D EndK D EndK K be the regular action defined by .x/y xy. W ! D D By Step 3, ˛ is trivial on the commutator group N 0, thus

.SL2 R/ x a x a N x ax a N .D/: 00 Â ¹ 7!  j 2 º0 D ¹ 7! j 2 0º Â In particular, for every a R the matrix 2 ! ! ! 1 ! 1 ! 0 1 1 a 0 1 1 a 1 a La WD 1 0 0 1 1 0 0 1 D a 1 a2 C belongs to .SL2 R/0, hence the matrix

2 3 3 ! 1 1 1 a a 2a a a Ma LaL1L L C C .SL2 R/00 WD a 1 D 2a a2 a3 2a4 1 a a4 2 C C belongs to .D/. With our assumption R 4, the inclusion of these matrices j j  1 Ma in .D/ will lead to a contradiction via some computation . We recall that M M 1 trace.M /id for every 2 2-matrix M with determinant 1. C D  If K has characteristic 2, then ! a a2 a a3 Ma id C C ; D C a2 a3 a a4 C C 1 . . . but when the chips are down we close the office door and compute with matrices like fury. (Irving Kaplansky) 716 T. Grundhöfer and the skew field .D/ contains all matrices 1 M 2 M 2 Ma M M 2 M 2 trace.Ma/id a 1 C a C C a D a 1 C a C C C ! 1 1 a2.a 1/2 D C 1 1 with a R. Choosing a R F2 we have a matrix of rank 1 in .D/. This 2 2 n ¤ ; is a contradiction, because all non-zero matrices in the skew field .D/ D are Š invertible. Now we assume that K does not have characteristic 2. For every a R the skew 2 field .D/ contains the matrix 1 1 3 Ma Ma .M a M a / trace.Ma/id trace.M a/id 4a id; C C D D hence a3 id .D/ by our assumption on the characteristic. Thus the skew field 2 .D/ contains all matrices ! 6 6 0 0 2.a 1/id Ma3 M a3 2a 6 6 C C D 2a 1 a 1 C 0 0  with a R. Putting a 1 we find that .D/ contains the nilpotent matrix 3 0 , 2 D 2 hence 3 0 and K has characteristic 3. For a R F3 we have a 1, D 2 n ¤ ; ¤ hence a6 1; then the matrix considered above has rank 1 and belongs to the ¤ skew field .D/, which is a contradiction.

Theorem 3.1 and the constructions in [4] yield the following result.

Corollary 3.3. There exist infinite nearfields of any characteristic that are wild, i.e. not Dickson nearfields.

Bibliography

[1] E. Artin, Algebraic Numbers and Algebraic Functions, Gordon and Breach, New York, 1967. [2] D. E. Cohen, Combinatorial : A Topological Approach, Cambridge University Press, Cambridge, 1989. [3] W. Dicks and M. J. Dunwoody, Groups Acting on Graphs, Cambridge University Press, Cambridge, 1989. [4] T. Grundhöfer, Free products and wild nearfields à la B. H. Neumann and , Forum Math. (2012), DOI 10.1515/forum-2012-0065. [5] T. Grundhöfer and H. Zassenhaus, A criterion for infinite non-Dickson nearfields of dimension two, Results Math. 15 (1989), 221–226. Five wild nearfields 717

[6] N. Jacobson, Structure of Rings, rev. ed, American Mathematical Society, Provi- dence, 1984. [7] H. Karzel, Unendliche Dicksonsche Fastkörper, Arch. Math. 16 (1965), 247–256. [8] J. Leitzel, M. Madan and C. Queen, Algebraic function fields with small class num- bers, J. Number Theory 7 (1975), 11–27. [9] R. E. MacRae, On unique factorization in certain rings of algebraic functions, J. Al- gebra 17 (1971), 243–261.

[10] A. W. Mason, Anomalous normal subgroups of SL2.KŒx/, Quart. J. Math. Oxford (2) 36 (1985), 345–358.

[11] A. W. Mason, Normal subgroups of SL2.kŒt/ with or without free quotients, J. Al- gebra 150 (1992), 281–295. [12] A. W. Mason, Serre’s generalization of Nagao’s theorem: An elementary approach, Trans. Amer. Math. Soc. 353 (2001), 749–767. [13] A. W. Mason and A. Schweizer, The minimum index of a non-congruence subgroup of SL2 over an arithmetic domain. II: The rank zero cases, J. Lond. Math. Soc. (2) 71 (2005), 53–68. [14] A. W. Mason and A. Schweizer, The stabilizers in a Drinfeld of the vertices of its Bruhat–Tits tree: An elementary approach, preprint (2013), http: //arxiv.org/abs/1203.3617. [15] H. Nagao, On GL.2; KŒx/, J. Inst. Polytech. Osaka City Univ. Ser. A 10 (1959), 117–121. [16] B. H. Neumann, Über ein gruppentheoretisch-arithmetisches Problem, Sitzungsbe- richte Preuss. Akad. Wiss. Berlin Phys.-Math. Klasse 1933 (1933), no. 7–10, 429– 444. [17] M. Newman, A note on modular groups, Proc. Amer. Math. Soc. 14 (1963), 124–125. [18] A. Y. Ol’shanskij and A. L. Shmel’kin, Infinite groups, in: Algebra IV, Encyclopaedia Math. Sci. 37, Springer-Verlag, Berlin (1993), 1–95. [19] D. Passman, Permutation Groups, Benjamin, New York, 1968, Dover, Mineola, 2012. [20] J.-P. Serre, Trees, Springer-Verlag, Berlin, 1980. [21] S. Takahashi, The fundamental domain of the tree of GL.2/ over the function field of an , Duke Math. J. 72 (1993), 85–97. [22] H. Wähling, Theorie der Fastkörper, Thales-Verlag, Essen, 1987. [23] H. Zassenhaus, Über endliche Fastkörper, Abh. Math. Semin. Univ. Hambg. 11 (1936), 187–220. [24] H. Zassenhaus, On Frobenius groups II: Universal completion of nearfields of finite degree over a field of reference, Results Math. 11 (1987), 317–358. 718 T. Grundhöfer

Received September 13, 2012.

Author information Theo Grundhöfer, Institut für Mathematik, Universität Würzburg, Emil-Fischer-Str. 30, 97074 Würzburg, Germany. E-mail: [email protected]