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Several types of solvable groups as automorphism groups of compact Riemann surfaces Andreas Schweizer Department of Mathematics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 305-701 South Korea e-mail: [email protected] Abstract Let X be a compact Riemann surface of genus g ≥ 2. Let Aut(X) be its group of automorphisms and G ⊆ Aut(X) a subgroup. Sharp upper bounds for |G| in terms of g are known if G belongs to certain classes of groups, e.g. solvable, supersolvable, nilpotent, metabelian, metacyclic, abelian, cyclic. We refine these results by finding similar bounds for groups of odd order that are of these types. We also add more types of solvable groups to that long list by establishing the optimal bounds for, among others, groups of order pmqn. Moreover, we show that Zomorrodian’s bound for p-groups G with p ≥ 5, 2p namely |G| ≤ p−3 (g − 1), actually holds for any group G for which p ≥ 5 is the smallest prime divisor of |G|. Mathematics Subject Classification (2010): primary 14H37; 30F10; sec- ondary 20F16 Key words: compact Riemann surface; automorphism group; group of odd order; solvable; supersolvable; nilpotent; metabelian; metacyclic; CLT group; (p,q)-group; smallest prime divisor 1. Introduction ′ arXiv:1701.00325v2 [math.CV] 4 Jul 2017 We write |G| for the order of a group G and G for the commutator group. Also Cn stands for a cyclic group of order n. In this paper X will always be a compact Riemann surface of genus g ≥ 2. Its full group of conformal automorphisms is denoted by Aut(X). It is a classical theorem by Hurwitz that then |Aut(X)| ≤ 84(g − 1). See for example [B, Theorem 3.17]. However, not every group whose order is divisible by 84 can occur in Hurwitz’s Theorem. For example, if g =7n +1, a group of order 84(g − 1) = 12 · 7n+1 must be solvable by the Sylow Theorems; and for solvable groups there are stronger bounds (see Theorem 2.2 (a) below). More generally, if G is a (not necessarily proper) subgroup of Aut(X) and G belongs to a certain type of groups, one is interested in having a bound on |G| in terms of g. Ideally one would like to have for any given type of groups a simple 1 function b(g) such that |G| ≤ b(g) except for finitely many (explicitly known) g, and that for infinitely many values of g there is a G of the desired type with |G| = b(g). Such a function b(g) is of course not guaranteed to exist. But for many interesting types of groups it does. From the rich literature we summarize the results that are relevant to this paper. TABLE 1 class of G upper bound exceptions source general 84(g − 1) none classical solvable 48(g − 1) none [Ch], [G1], [G2] supersolvable 18(g − 1) for g =2 [Z3], [GMl], [Z4] nilpotent 16(g − 1) none [Z1] metabelian 16(g − 1) for g =2, 3, 5 [ChP], [G2], [G3] metacyclic 12(g − 1) for g =2 [Sch2] Z-group 10(g − 1) for g =2, 3 [Sch2] |G| is square-free 10(g − 1) for g =3 [Sch2] abelian 4g +4 none classical, see also [Ml], [G1] cyclic 4g +2 none classical, see also [Ha], [G1], [N] In Theorem 2.2 in the next section we will report more details on some of the groups. Here we only mention that the first proof for metabelian groups is in [ChP]. But the last theorem of [G2] points out that a metabelian group of order 24 for g = 2, covered by Γ(0; 2, 4, 6), had been overlooked. See also [G3] for a detailed de- scription of the metabelian G with |G| = 16(g − 1) and some corrections concerning their possible orders. Bounds and more information if G is a p-group were worked out in [Z2]; see Theorem 2.3. More recent results in [W2] and [MZ2] include the optimal bound b(g) for subgroups G of odd order in Aut(X), which we recall in Theorem 2.4. In this paper we present several new results of a similar nature. In Section 5 we refine the known results by determining for each type of group in Table 1 the sharp bound b(g) if G ⊆ Aut(X) is an odd order subgroup of that type. It turns out that odd order supersolvable G of maximal possible order are subject to much stronger restrictions than the other types (see Theorem 5.1). The idea of considering groups of odd order can be generalized to prescribing the smallest prime divisor p of |G|. We do this already in Section 4, before treating odd order groups, for reasons having to do with the logical dependence of the proofs. It 2 turns out that for p ≥ 5 the bounds for the first 8 types of groups in Table 1 all collapse to the same bound, namely the one for p-groups from Theorem 2.3 (c). For a prime number p denote by G(p) the class of all finite groups whose orders are not divisible by any primes smaller than p. So G(2) denotes all finite groups and G(3) all groups of odd order. Omitting some of the details, the following big picture emerges. The bound for p-groups from Theorem 2.3 is also the bound for nilpotent groups inside the class G(p), and any nilpotent group in G(p) that reaches this bound must be a p-group. Moreover, with the exception of 6 individual groups (3 in G(2) and 3 in G(3)), this bound also is the sharp bound for metabelian groups in G(p). If p ≥ 3 and q ≡ 1 mod p are primes, we can even find infinitely many metabelian, supersolvable (p, q)-groups that attain this bound. So in Section 6 we treat yet another type of groups that are known to be solvable, namely groups of order pmqn where p < q are primes. We use the results from the previous sections to get optimal bounds on |G| for G ⊆ Aut(X) with |G| = pmqn, in general and if p and/or q is prescribed. In Section 7 we get similar results for two (probably less important) types of groups that are lying strictly between the supersolvable and the solvable ones, namely groups with nilpotent commutator group, and groups in which the elements of odd order form a (necessarily normal) subgroup. In Section 8 we derive sharp upper bounds on |G| for CLT groups of odd order. CLT groups are another type of groups between the supersolvable and the solvable ones. They are defined in elementary terms by the condition that for every divisor d of |G| there is a subgroup of order d. But CLT groups are not at all well-behaved. See Section 8 for details. This makes them a very unwieldy object to handle and explains perhaps why we only get partial results for the size of CLT groups in general. 2. Known results Let X be a compact Riemann surface of genus g ≥ 2 and let G be a subgroup of Aut(X). The key tool to get information on G is that G ⊆ Aut(X) is covered by a Fuchsian group Γ = Γ(h; m1,...,mr). This means that Γ is discretely embedded into Aut(U), where U is the complex upper halfplane, and that there is a torsion- free normal subgroup K of Γ with Γ/K ∼= G such that U/K ∼= X is the universal covering. Moreover, h is the genus of X/G ∼= U/Γ. See Section 3 in Chapter 1 of [B] for more background and references. Theorem 2.1. If G is covered by a Fuchsian group Γ(h; m1, m2,...,mr), then 2 |G| = (g − 1). 2h − 2+ Pr (1 − 1 ) i=1 mi Proof. [B, Theorem 3.5 and p.15]. 3 In particular, if |G| > 4(g − 1), then h = 0. In practically all cases we are in- terested in, Γ even is a triangle group, i.e. h = 0 and r = 3. Since we are interested in solvable groups G ⊆ Aut(X), the biggest abelian quotient G/G′ is very important for us. It must be a quotient of Γ/Γ′, the structure of which can be easily read off (at least if h = 0) from the generators and relations ∼ m1 m2 mr Γ(0; m1,...,mr) = hx1,...,xr | x1 = x2 = ... = xr = x1x2 ··· xr =1i. For frequent use throughout the paper we take from Table 4.1 in [GMl] all pos- sible orders |G| ≥ 18(g − 1) for solvable groups G ⊆ Aut(X) together with the corresponding triangle groups Γ. TABLE 2 |G| Γ Γ/Γ′ Γ′ 48(g − 1) Γ(0;2, 3, 8) C2 Γ(0;3, 3, 4) 40(g − 1) Γ(0;2, 4, 5) C2 Γ(0;5, 5, 2) 36(g − 1) Γ(0;2, 3, 9) C3 Γ(0;2, 2, 2, 3) 30(g − 1) Γ(0;2, 3, 10) C2 Γ(0;3, 3, 5) 24(g − 1) Γ(0;2, 3, 12) C6 Γ(1; 2) 24(g − 1) Γ(0;2, 4, 6) C2 × C2 Γ(0;2, 2, 3, 3) 24(g − 1) Γ(0;3, 3, 4) C3 Γ(0;4, 4, 4) 21(g − 1) Γ(0;2, 3, 14) C2 Γ(0;3, 3, 7) 20(g − 1) Γ(0;2, 3, 15) C3 Γ(0;2, 2, 2, 5) 20(g − 1) Γ(0;2, 5, 5) C5 Γ(0;2, 2, 2, 2, 2) 96 5 (g − 1) Γ(0;2, 3, 16) C2 Γ(0;3, 3, 8) 56 3 (g − 1) Γ(0;2, 4, 7) C2 Γ(0;7, 7, 2) 18(g − 1) Γ(0;2, 3, 18) C6 Γ(1; 3) Actually, Table 4.1 in [GMl] contains four more triangle groups with |G| ≥ 18(g−1), namely Γ = Γ(0;2, 3,p) with p =7, 11, 13, 17.
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