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Five Wild Nearfields J. Group 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 subgroups 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 special linear group SL2 Fp is solvable. As a consequence, the 2 ¹ º commutator subgroup 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 permutation group 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 kernel, i.e. a sharply transitive normal subgroup. 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 integer. 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 symmetric group S3 for p 2 and isomorphic to the D alternating group 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 order 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 image 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.
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