Hurwitz groups with given centre

Marston Conder

Department of Mathematics

University of Auckland

Private Bag 92019 Auckland, New Zealand

         conder@math.auckland.ac.nz

Dedicated to Peter Neumann on the occasion of his 60th birthday

Abstract

A Hurwitz group is any non-trivial finite group which can be (2, 3, 7)-generated, that is, generated by elements x and y satisfying the relations x^{2} = y^{3} = (xy)^{7} = 1. In this short paper a complete answer is given to a 1965 question by John Leech, showing that the centre of a Hurwitz group can be any given finite abelian group. The proof is based on a recent theorem of Lucchini, Tamburini and Wilson which states that the special linear group SL_{n}(q) is a Hurwitz group for every integer n ≥ 287 and every prime-power q.

Mathematics Subject Classification: 20F05 (primary), 57M05 (secondary)

1 Introduction

The (2, 3, 7) triangle group is the abstract group Δ with presentation

Δ = ⟨ x, y, z | x^{2} = y^{3} = z^{7} = xyz = 1 ⟩.

A theorem of Hurwitz (1893) states that any compact Riemann surface with genus g > 1 admits at most 84(g − 1) conformal automorphisms, that is, homeomorphisms of the surface onto itself which preserve the local structure. Further, any such surface X with the maximum number of automorphisms must be uniformized by a normal subgroup N of the (2, 3, 7) triangle group Δ, for the latter is the Fuchsian group with fundamental region of smallest hyperbolic area, and the conformal automorphism group of X is then isomorphic to the quotient group Δ/N. Conversely, any non-trivial finite quotient G of Δ gives rise to a compact Riemann surface (of genus |G|/84 + 1) with the maximum possible number of automorphisms, and G as its automorphism group. For these reasons, any non-trivial finite quotient of Δ is called a Hurwitz group. Further details of their significance may be found in [4].

Every Hurwitz group G is perfect (that is, has trivial abelianisation G/G′), and so it is natural to look among the finite nonabelian simple groups for examples of Hurwitz groups. A search was initiated by Macbeath in the 1960s when he proved in [11] that the simple group PSL_{2}(q) is a Hurwitz group precisely when q = 7, or q is a prime congruent to ±1 modulo 7, or q = p^{3} where p is a prime congruent to ±2 or ±3 modulo 7. Graham Higman developed a method of composing coset diagrams to prove that all but finitely many alternating groups are Hurwitz, and the author of this paper took this further to prove in [5] that A_{n} is Hurwitz far all n ≥ 168 (and many smaller values of n as well). Many other examples are now known, including G_{2}(q) for all q > 4, certain Ree groups, and some of the sporadic simple groups; a recent summary was given in by Jones in [7].

In a very recent (and amazing) piece of work, Robert Wilson applied clever computational techniques to show that the Monster M is a Hurwitz group, using its 196882-dimensional irreducible representation over GF(2); see [12]. In Wilson’s words, this “demonstrates that, against all reasonable expectations, it is now possible to do practical calculations in this enormous group”. In particular, it is now known that exactly 12 of the sporadic finite simple groups are Hurwitz groups, namely J_{1}, J_{2}, He, Ru, Co_{3}, Fi_{22}, HN, Ly, Th, J_{4}, Fi′_{24} and M (see [6] and other references in [7] and [12]).

Also Lucchini, Tamburini and John Wilson have now shown that most finite simple classical groups of sufficiently large dimension are Hurwitz groups. In particular, in [10] they adapted the methods used by the author in [5] to prove that the special linear group SL_{n}(q) is Hurwitz for every n ≥ 287 and every prime-power q. We will subsequently call this the Lucchini/Tamburini/Wilson theorem. Further, in [9] Lucchini and Tamburini showed that the classical groups Sp_{2n}(q), Ω_{2n}^{+}(q) and SU_{2n}(q) are Hurwitz for every n ≥ 371 and every prime-power q, and that Ω_{2n+7}(q) and SU_{2n+7}(q) are Hurwitz for every n ≥ 371 and every odd prime-power q. It follows that simple (projective) quotients of these groups are all Hurwitz.

Turning now more towards the subject of this paper, we consider the question of what centres are possible in finite quotients of the (2, 3, 7) triangle group. This question was raised in 1965 by Leech [8], who later produced two infinite families of Hurwitz groups with centres of orders 2 and 4, from extensions of 7-generator nilpotent groups by PSL_{2}(8). In [2] the author of this paper used a similar method to proved the existence of infinitely many Hurwitz groups with a centre of order 3, and in [3] constructed a family of central products of 2-dimensional special linear groups to show that the centre of a Hurwitz group could be an elementary abelian 2-group of arbitrarily large order.

In this paper we use the Lucchini/Tamburini/Wilson theorem (Corollary 1 of the main theorem of [10]) to prove that the centre of a Hurwitz group can be any finite abelian group. In fact we prove the following:

Theorem Given any finite abelian group A, there exist infinitely many Hurwitz groups G such that the centre Z(G) of G is isomorphic to A.

2 Proof of Theorem

The crux of the proof is quite simple: take a direct product of appropriately chosen special linear Hurwitz groups.

Let A be any finite abelian group. Then A is isomorphic to a direct product of cyclic groups, say C_{m_{1}} × C_{m_{2}} × … × C_{m_{s}}, of orders m_{i} ≥ 2 for 1 ≤ i ≤ s. Now choose any prime p which does not divide |A| = m_{1}m_{2} … m_{s}. Then p is congruent to a unit in the ring of integers modulo |A| and so there exists a positive integer e for which p^{e} ≡ 1 mod |A|. (For example, take e = ϕ(|A|).)

Let q = p^{e}. Then for 1 ≤ i ≤ s we know q ≡ 1 modulo m_{i} and therefore q − 1 = l_{i}m_{i} for some integer l_{i}. Further, if k_{i} is any positive integer coprime to l_{i}, then letting n_{i} = k_{i}m_{i} we have gcd(n_{i}, q − 1) = gcd(k_{i}m_{i}, l_{i}m_{i}) = m_{i}. In particular, there are infinitely many possibilities for each n_{i}, and all can be chosen such that the n_{i} are mutually distinct, and n_{i} ≥ 287 for 1 ≤ i ≤ s.

Next let H_{i} = SL_{n_{i}}(q) for 1 ≤ i ≤ s. By the Lucchini/Tamburini/Wilson theorem, each H_{i} is a Hurwitz group, so can be generated by two elements X_{i} and Y_{i} satisfying the relations X_{i}^{2} = Y_{i}^{3} = (X_{i}Y_{i})^{7} = 1. Note that the centre Z(H_{i}) of H_{i} consists of all the scalar matrices λI_{n_{i}} of determinant 1, and hence is cyclic of order gcd(n_{i}, q −1) = m_{i}. Also H_{i}/Z(H_{i}) is isomorphic to the nonabelian simple group PSL_{n_{i}}(q), for 1 ≤ i ≤ s, and because the n_{i} were chosen to be distinct, H_{i}/Z(H_{i}) ≅ H̸_{j}/Z(H_{j}) whenever i ≠ j.

Define G to be the direct product Π_{1≤i≤s} H_{i} of these special linear Hurwitz groups, and let X = X_{1}X_{2} … X_{s} and Y = Y_{1}Y_{2} … Y_{s}. As the X_{i} commute with each other, X has order 2, and similarly Y has order 3. Also the product XY = X_{1}Y_{1}X_{2}Y_{2} … X_{s}Y_{s} has order 7. Hence the subgroup H generated by X and Y is a Hurwitz group.

We claim that H = G. To see this, note first that the natural projection π_{i} : G → H_{i} maps (X, Y) to (X_{i}, Y_{i}) and therefore maps H onto H_{i}, for 1 ≤ i ≤ s. In particular, H has distinct composition factors isomorphic to PSL_{n_{i}}(q) for 1 ≤ i ≤ s. Now for the purposes of induction, let us assume that the natural homomorphism ξ_{i} : G → Π_{j≠i} H_{j} (obtained by suppressing the ith term of the product) maps H onto Π_{j≠i} H_{j}. The kernel of ξ_{i} is isomorphic to a subgroup G_{i} of H_{i} = SL_{n_{i}}(q) having PSL_{n_{i}}(q) as a composition factor, so that SL_{n_{i}}(q) = Z_{i}G_{i} where Z_{i} is the centre of SL_{n_{i}}(q). But now Z_{i} ∩ G_{i} is a central subgroup of SL_{n_{i}}(q), and SL_{n_{i}}(q)/(Z_{i} ∩ G_{i}) is isomorphic to the direct product Z_{i}/(Z_{i} ∩ G_{i}) × G_{i}/(Z_{i} ∩ G_{i}), which forces Z_{i}/(Z_{i} ∩ G_{i}) to be trivial (because SL_{n_{i}}(q) is perfect). Hence Z_{i} ⊆ Z_{i} ∩ G_{i} ⊆ G_{i}, and therefore G_{i} = Z_{i}G_{i} = SL_{n_{i}}(q) = H_{i}. As this argument holds for 1 ≤ i ≤ s, it follows by induction on the number of factors that H = Π_{1≤i≤s} H_{i} = G.

The centre of G is clearly Π_{1≤i≤s} Z(H_{i}), which is a direct product of cyclic groups of orders m_{i} for 1 ≤ i ≤ s; that is, Z(G) ≅ A as required.

Note: There are infinitely many possibilities for the prime p, and hence also for the prime-power q = p^{e}, and also infinitely many possibilities for the ranks n_{1}, n_{2}, … n_{s}. Moreover, other constructions are possible. For example, by Dirichlet’s Theorem on primes in arithmetic progression (see [1]), there are infinitely many possible choices of distinct primes p_{1}, p_{2}, …, p_{s} such that p_{i} ≡ 1 modulo m_{i} for 1 ≤ i ≤ s, and accordingly we may take G to be the direct product of special linear Hurwitz groups H_{i} = SL_{n_{i}}(p_{i}) with n_{i} divisible by m_{i}, and then factor out however much of the centre of this product is surplus to requirements. In this case each Z(H_{i}) contains a central subgroup K_{i} of H_{i} of index m_{i} in Z(H_{i}), and hence Z(G) contains a central subgroup K = Π_{1≤i≤s} K_{i} of G of index m_{1}m_{2} … m_{s} in Z(G), such that the Hurwitz group G/K has centre Z(G/K) ≅ Z(G)/K ≅ Π_{1≤i≤s} C_{m_{i}} ≅ A.

Acknowledgement The author is grateful to the N.Z. Marsden Fund for financial support (grant number UOA 810), and to the referee for helpful comments which led to an improved version of an early draft of this paper.

References

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[8] J. Leech, Generators for certain normal subgroups of (2, 3, 7), Proc. Cambridge Philos. Soc. 61 (1965), 321–332.

[9] A. Lucchini and M.C. Tamburini, Classical groups of large rank as Hurwitz groups, J. Algebra 219 (1999), 531–546.

[10] A. Lucchini, M.C. Tamburini and J.S. Wilson, Hurwitz groups of large rank, J. London Math. Soc. (2) 61 (2000), 81–92.

[11] A.M. Macbeath, Generators of the linear fractional groups, Number theory (Proc. Sympos. Pure Math., Vol XII, 1967), Amer. Math. Soc., 1969, pp. 14–32.

[12] R.A. Wilson, The Monster is a Hurwitz group, preprint.

Author’s address: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand

Author’s e-mail address: conder@math.auckland.ac.nz