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THE STRONG SYMMETRIC GENUS and GENERALIZED SYMMETRIC GROUPS of TYPE G(N, 3)

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THE STRONG SYMMETRIC GENUS and GENERALIZED SYMMETRIC GROUPS of TYPE G(N, 3) THE STRONG SYMMETRIC GENUS AND GENERALIZED SYMMETRIC GROUPS OF TYPE G(n, 3) MICHAEL A. JACKSON Abstract. The generalized symmetric groups are defined to be G(n, m) = Zm o Σn where n, m ∈ Z+. The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation preserving automorphisms. The present work extends work on the strong symmetric genus by Marston Conder, who studied the symmetric groups, which are G(n, 1), and by the author, who studied the hyperoctahedral groups, which are G(n, 2). We give results for the strong symmetric genus of the groups of type G(n, 3). Grove City College, 100 Campus Drive, Grove City, PA 16127 [email protected]. 1 1. Introduction The strong symmetric genus of the finite group G is the smallest genus of a compact orientable surface on which G acts as a group of orientation preserving symmetries and is denoted σ0(G). The strong symmetric genus has a long history beginning with Burnside [1]. For more general information see [10, Chapter 6]. The strong symmetric genus of many groups are known: the alternating and symmetric groups [2, 3, 4], the hyperoctahedral groups [12], the remaining finite Coxeter groups [11], the groups P SL2(q) [8, 9] and SL2(q) [15], as well as the sporadic finite simple groups [5, 16, 17, 18]. The generalized symmetric groups G(n, m) are defined for n > 1 and m ≥ 1 to be the wreath product Zm o Σn. We notice that the traditional symmetric groups are the generalized symmetric groups of type G(n, 1), and the hyperoctahedral groups are the generalized symmetric groups of type G(n, 2). Since the strong symmetric genus has been found for these two families of generalized symmetric groups, in this paper we will find the strong symmetric genus for the next family of generalized symmetric groups: type G(n, 3). We will prove the following two theorems and a corollary: Theorem 1. For all n > 33, the generalized symmetric group G(n, 3) is a quotient of the triangle group T (2, 3, 12) = hx, y, z|x2 = y3 = z12 = xyz = 1i. 2 Corollary 2. For all n > 33, the generalized symmetric group G(n, 3) has strong n!3n n!2n−1 symmetric genus 24 + 1 = 8 + 1. Theorem 3. For 2 ≤ n ≤ 33, the strong symmetric genus of the generalized symmetric group G(n, 3) is given in Table I along with the appropriate triangle groups. The result for G(3, 3) was found by Dustin Pierce, a student at King College. I would like to thank the editor and referee for proving many helpful comments and suggestions which have allowed this paper to take its present form. 2. Subgroups and generators of the generalized symmetric groups For the present time, we will fix n ≥ 3 as well as fix a prime p ≥ 3, and we will look at the generalized symmetric group G(n, p). As previously stated, G(n, p) is the wreath product Zp o Σn. G(n, p) is also as the group of all n × n generalized permutation matrices with nonzero entries in a multiplicative cyclic group Cp. For an element of G(n, p), we will write an ordered pair [σ, b] where σ is an element of Σn and n b is an n-tuple of elements in Zp (i.e. an element of (Zp) ). The multiplication then becomes [σ, b]·[τ, c] = [σ·τ, τ −1(b)+c] where addition in the n-tuple is the operation in the group Zp performed in each position. Recall that G(n, p) is a semidirect product n of (Zp) by Σn. n Definition 4. For an element b ∈ (Zp) , we say that b is divisible by p if the sum of the terms of b add to the identity of Zp. If we fix a generator of Zp, we can 3 n define a group homomorphism Σ : (Zp) → Zp where the image of b is the sum of the elements in the n-tuple b. Notice that Σ is invariant under permutations of b, and that b is divisible by p if and only if Σ(b) = 0. We now present a proposition that is well known (see [6, 13]). Proposition 5. For n ≥ 5, let G be a subgroup of G(n, p) with π(G) = Σn; then n−1 n G is a split extension of Σn by one of the following: 1, Zp, (Zp) , or (Zp) . In the first two cases, G is isomorphic to Σn and Zp × Σn respectively. In the third case, G = {[σ, b] ∈ G(n, p)|b is divisible by p}, while G = G(n, p) in the last case. We finish this section with a proposition that constructs generators of G(n, p) from generators of Σn Proposition 6. Suppose σ and τ generate Σn such that o(σ) = 2 and o(τ) = p for some odd prime p. Futhermore, assume that τ fixes an element i ∈ {1, 2, . , n}; and in the case where n ≡ 1(mod p), σ · τ has a cycle of length not divisible by p and not containing i. Let l be the length of the cycle in σ · τ that contains i. Let b = (0,..., 0) and c = (0,..., 0, 1, 0,..., 0), where the 1 is in the ith position, be n elements in an additive group (Zp) . Under these conditions, [σ, b] and [τ, c] generate G(n, p). In addition the elements [σ, b], [τ, c] and [σ · τ, τ −1(b) + c] have orders 2, p, and gcd(o(σ · τ), p · l) respectively. 4 Proof. Let G = h[σ, b], [τ, c]i ⊂ G(n, p). Notice that any section s :Σn → G(n, p) takes α ∈ Sn to [α, a] where a is divisible by p. Clearly [τ, c] can not be in the image of any section homomorphism and so G is not isomorphic to Σn. Similarly since [τ, c] n−1 is not divisible by p, G is not a split extension of Σn by (Zp) by Proposition 5. Suppose that G is a split extension of Σn by Zp. By Proposition 5, for some section homorphism s :Σn → G(n, p), G = Z(G(n, p)) × s(Σn). Letting [1, z] be a generator of Z(G(n, p)), there is a [1, d] ∈ G(n, p) such that for each [α, a] ∈ G, −1 −1 n τ (d)+d = a+k z in (Zp) for some scalar 0 ≤ k < p. If τ fixes some element j that −1 −1 n is different from i, then τ (d)+d 6= c+k z for all d ∈ (Zp) ; therefore n ≡ 1(mod p). By hypothesis we can assume that σ · τ has a cycle χ of length not divisible by p and not containing i. Notice that any subset of (σ · τ)−1(d) + d−1 is divisible by p if it −1 is closed under permutation by σ · τ. Since [σ · τ, τ (b) + c] ∈ Z(G(n, p)) × s(Σn), τ −1(b) + c + k z = c + k z = (σ · τ)−1(d) + d−1 for some 0 ≤ k < p. Using the portion of (σ · τ)−1(d) + d−1 permuted by χ, we see that k = 0, because this portion of k · z must be divisible by p. Now considering the entirety of (σ · τ)−1(d) + d−1, we see that c + k z = c must be divisible by p, which is not possible. Applying Proposition 5, the argument above implies that G = G(n, p). The results −1 about the orders of [σ, b], [τ, c] and [σ·τ, τ (b)+c] follow from an easy calculation. 5 3. A generalization of Singerman’s Lemma We begin section 3 with a result about the order of the generators of generalized symmetric groups. Lemma 7. Let G = G(n, p) be a generalized symmetric group with n > 3 and p > 2 a prime. If x1, x2, x3, . , xn are generators of G, then at least two of the orders of x1, x2, x3, . , xn and their product x1x2x3 ··· xn are divisible by p. Proof. Recall that G(n, p) is defined as a split extension as follows: n i π 1 → (Zp) → G(n, p) → Σn → 1. In addition we have s :Σn → G(n, p) with π ◦ s = idΣn . For each xi, we let −1 −1 −1 −1 zi = s◦π(xi)·xi . Since π(zi) = π◦s◦π(xi)·π(xi ) = π(xi)·π(xi ) = π(xi ·xi ) = 1, it is clear that zi is in the kernel of π. Thus zi is in the image of i. Since i is injective, let yi be the unique preimage of zi under i. Recall the group homomorphism n Σ:(Zp) → Zp. Since the xi’s generate G(n, p), at least one Σ(yi) must be a generator of Zp. If Σ(yi) is a generator, then the order of xi is divisible by p. Also if only one of the Σ(yi)’s is a generator of Zp, then Σ(y1y2 ··· yn) will also be a generator of Zp. Thus in this case, the order of x1x2x3 ··· xn is divisible by p. Theorem 8 (A generalization of Singerman’s Lemma (See [14])). Let G = G(n, 3) be a generalized symmetric group with n > 3. If |G| > 6(σ0(G) − 1), then G has a 6 (p, q, r) generating pair with 1 1 1 1 σ0(G) = 1 + |G| · (1 − − − ).
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