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Introduction Known Results Small Strong Symmetric Genus Quotient (2, 3, r ) for r Relatively Prime to 6 There is a group of every possible small strong symmetric genus quotient Michael A. Jackson Department of Mathematics - Grove City College majackson@gcc.edu 2011 Zassenhaus Group Theory Conference Towson University May 27, 2011 includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Small Strong Symmetric Genus Quotient (2, 3, r ) for r Relatively Prime to 6 Outline 1 Introduction Strong Symmetric Genus Quotient Minimal Generating Pairs Singerman’s Lemma 2 Known Results Strong Symmetric Genus Projective Special Linear Groups Generalized Symmetric Groups 3 Small Strong Symmetric Genus Quotient Minimal Generating Pairs that are not (2, 3, r ) (2, 3, r ) for r not relatively prime to 6 4 (2, 3, r ) for r Relatively Prime to 6 Square Free r General r includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Quotient Known Results Minimal Generating Pairs Small Strong Symmetric Genus Quotient Singerman’s Lemma (2, 3, r ) for r Relatively Prime to 6 Strong Symmetric Genus Deﬁnition Given a ﬁnite group G, the smallest genus of any closed orientable topological surface on which G acts faithfully as a group of orientation preserving symmetries is called the strong symmetric genus of G. The strong symmetric genus of the group G is denoted σ0 (G). By a result of Hurwitz [1893], if σ0 (G) > 1 for a ﬁnite group G, then σ0 (G) ≥ 1 + |G| . 84 includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Quotient Known Results Minimal Generating Pairs Small Strong Symmetric Genus Quotient Singerman’s Lemma (2, 3, r ) for r Relatively Prime to 6 Strong Symmetric Genus Quotient Deﬁnition Given a ﬁnite group G with σ0 (G) > 1, we deﬁne the strong symmetric genus quotient to be σ0 (G) − 1 σQ (G) = . |G| By the result of Hurwitz [1893], for any ﬁnite group G with σ0 (G) > 1, σQ (G) ≥ 84 . 1 includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Quotient Known Results Minimal Generating Pairs Small Strong Symmetric Genus Quotient Singerman’s Lemma (2, 3, r ) for r Relatively Prime to 6 Generators and the Riemann-Hurwitz Equation If a ﬁnite group G has generators x and y of orders p and q respectively with xy having the order r , then we say that (x, y ) is a (p, q, r ) generating pair of G. The existance of a (p, q, r ) generating pair gives a faithful orientation preserving action of the group G on a surface S. The genus of the surface S is found from the Riemann-Hurwitz formula: |G| 1 1 1 genus(S) = 1 + (1 − − − ). 2 p q r includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Quotient Known Results Minimal Generating Pairs Small Strong Symmetric Genus Quotient Singerman’s Lemma (2, 3, r ) for r Relatively Prime to 6 Minimal Generating Pairs A (p, q, r ) generating pair of G is called a minimal generating pair if no generating pair for the group G gives an action on a surface of smaller genus. Recall that we are only considering groups with σ0 (G) > 1. In these cases, any generating pair will be a (p, q, r ) generating pair with p + q + 1 < 1. 1 1 r With these assumptions a (p, q, r ) generating pair is minimal if 1 1 1 p + q + r is the largest of any generating pair. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Quotient Known Results Minimal Generating Pairs Small Strong Symmetric Genus Quotient Singerman’s Lemma (2, 3, r ) for r Relatively Prime to 6 A Lemma by Singerman Lemma (Singerman) Let G be a ﬁnite group such that σ0 (G) > 1. If |G| > 12(σ0 (G) − 1), then G has a (p, q, r ) generating pair with 1 1 1 1 σ0 (G) = 1 + |G| · (1 − − − ). 2 p q r Singerman’s Lemma implies that if G has a minimal (p, q, r ) 1 1 generating pair such that p + q + 1 ≥ 5 , then the strong r 6 symmetric genus is given by this generating pair. Since σ0 (G) > 1, we know that 1 p + 1 q + 1 r < 1. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Quotient Known Results Minimal Generating Pairs Small Strong Symmetric Genus Quotient Singerman’s Lemma (2, 3, r ) for r Relatively Prime to 6 A Lemma by Singerman Lemma (Singerman) Let G be a ﬁnite group such that σ0 (G) > 1. If σQ (G) < 1 12 , then G has a (p, q, r ) generating pair with 1 1 1 1 σQ (G) = (1 − − − ). 2 p q r Singerman’s Lemma implies that if G has a minimal (p, q, r ) 1 1 generating pair such that p + q + 1 ≥ 5 , then the strong r 6 symmetric genus (and strong symmetric genus quotient) is given by this generating pair. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Quotient Known Results Minimal Generating Pairs Small Strong Symmetric Genus Quotient Singerman’s Lemma (2, 3, r ) for r Relatively Prime to 6 More on Singerman’s Lemma By Singerman’s Lemma the only strong symmetric genus 1 quotients that can be less than 12 are those that are given by 1 1 1 1 2 (1 − p − q − r ) for a minimal (p, q, r ) generating pair with 5 6 < 1 p + 1 q + 1 r < 1. What triples of numbers (p, q, r ) ﬁt this requirement: (2, 3, r ) for any r . (2, 4, r ) for 5 ≤ r ≤ 11. (3, 3, r ) for r = 4 or r = 5. Notice that all but ﬁnitely many such triples are of the form (2, 3, r ). includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Known Results Projective Special Linear Groups Small Strong Symmetric Genus Quotient Generalized Symmetric Groups (2, 3, r ) for r Relatively Prime to 6 Strong Symmpetric Genus May and Zimmerman (2003) have shown that there is a group of every strong symmetric genus. They used the groups Zk × Dn . These groups have strong symmetric genus quotients larger than 1 12 . All groups G such that σ0 (G) ≤ 4 are known. (See Broughton, 1991; May and Zimmerman, 2000 and 2005) Recently all groups of strong symmetric genus up to 25 were found during a summer undergraduate research program. The search was conducted using using GAP’s small group library. (Fieldsteel, Lindberg, London, Tran, and Xu, 2008) (2008 Arizona Summer Program on Computational Group Theory) includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Known Results Projective Special Linear Groups Small Strong Symmetric Genus Quotient Generalized Symmetric Groups (2, 3, r ) for r Relatively Prime to 6 PSL2 (p) Glover and Sjerve (1985) showed that the group PSL2 (p) for a prime p has a minimal (2, 3, d) generating pair if p ≥ 13, p ±1 mod 5, and p ±1 mod 8, p−1 p+1 where d is the minimal integer at least 7 such that d| 2 or d| 2 . Using the results of Glover and Sjerve and Dirichlet’s theorem on primes in arithmetic progressions we can show that for any prime r ≥ 7 there is a prime p such that PSL2 (p) has a minimal (2, 3, r ) generating pair. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Known Results Projective Special Linear Groups Small Strong Symmetric Genus Quotient Generalized Symmetric Groups (2, 3, r ) for r Relatively Prime to 6 PSL2 (pn ) Glover and Sjerve (1987) also showed that the group PSL2 (pn ) for the prime p has a minimal (2, 3, d) generating pair if p = 2 and n ≥ 3, d is the minimal integer at least 7 such that d|2n − 1 or d|2n + 1 but d does not divide 2m − 1 or 2m + 1 for any m|n with 1 ≤ m < n. if p > 2 and n = 4, 5 of n ≥ 7, −1 n n d is the minimal integer at least 7 such that d| p 2 or d| p 2 , +1 pm −1 pm +1 d does not divide any of the integers 2 or 2 , where m|n and 1 ≤ m < n and d does nod divide any of the integers pm − 1 or pm + 1, where 2m|n and 1 ≤ 2m ≤ n. Using these results of Glover and Sjerve we can show that for any prime r ≥ 5 and n ≥ 2 there is a prime p > 2 such that n−1 PSL2 (pr (r −1)/2 ) has a minimal (2, 3, r n ) generating pair. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Known Results Projective Special Linear Groups Small Strong Symmetric Genus Quotient Generalized Symmetric Groups (2, 3, r ) for r Relatively Prime to 6 Generalized Symmetric Groups G(n, m) = Zm Sn for n > 1 and m ≥ 1. G(n, m) is the smallest group of n × n matrices containing both the permutation matrices and the diagonal matrices with entries in a multiplicative cyclic group of size m. G(n, 1) is the traditional symmetric group Sn . G(n, 2) is the hyperoctahedral group Bn . The strong symmetric genus has been found for the groups: G(n, 1) [Conder, 1980] G(n, 2) and G(n, 3) [J, 2004 and 2008] G(3, m), G(4, m) and G(5, m) [Ginter, Johnson, McNamara, 2008] includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Known Results Projective Special Linear Groups Small Strong Symmetric Genus Quotient Generalized Symmetric Groups (2, 3, r ) for r Relatively Prime to 6 D-type Generalized Symmetric Groups D(n, m) = (Zm )n−1 θ Sn for n > 1 and m ≥ 2. D(n, m) G(n, m) with an index of m. D(n, m) is the smallest group of n × n matrices containing both the permutation matrices and the diagonal matrices of determinant 1 with entries in a multiplicative cyclic group of size m. D(n, 2) is the Coxeter group Dn . The strong symmetric genus has been found for the groups D(n, 2) [J, 2007] includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Known Results Projective Special Linear Groups Small Strong Symmetric Genus Quotient Generalized Symmetric Groups (2, 3, r ) for r Relatively Prime to 6 AD-type Generalized Symmetric Groups AD(n, m) = (Zm )n−1 θ An for n > 1 and m ≥ 2. AD(n, m) D(n, m) with an index of 2. AD(n, m) G(n, m) with an index of 2m. AD(n, m) is the group of n × n matrices containing the permutation matrices of determinant 1 and the diagonal matrices of determinant 1 with entries in a multiplicative cyclic group of size m. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Strong Symmetric Genus Known Results Projective Special Linear Groups Small Strong Symmetric Genus Quotient Generalized Symmetric Groups (2, 3, r ) for r Relatively Prime to 6 Subgroups of Generalized Symmetric Group The strong symmetric genus of both D-type generalized symmetric groups and both AD-type generalized symmetric groups when n is 3, 4 or 5 have been found [Bowser, Partridge, Rodgers; 2009]. The following results are useful in this talk: When 3 m, D(3, m) has a minimal (2, 3, 2m) generating pair. When m is odd, AD(4, m) has a minimal (2, 3, 3m) generating pair. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Minimal Generating Pairs that are not (2, 3, r ) Small Strong Symmetric Genus Quotient (2, 3, r ) for r not relatively prime to 6 (2, 3, r ) for r Relatively Prime to 6 (2, 4, r ) Now we will show that there is a group giving each possible small symmetric genus quotient. We will begin with the minimal generating pairs of the form (2, 4, r ). Min. Gen. Pair example group σQ (G) Reference Conder, Wilson (2, 4, 5) J3 or Suz 1/40 and Woldar, 1992 (2, 4, 6) Bn for n ≥ 9 1/24 J, 2004 (2, 4, 7) S8 3/56 Conder, 1980 (2, 4, 8) B8 1/16 J, 2004 Bowser, Partridge (2, 4, 9) D(4, 3) 5/72 and Rodgers, 2009 (2, 4, 10) B5 3/40 J, 2004 (2, 4, 11) M11 7/88 Woldar, 1990 Bowser, Partridge (2, 4, 12) D(5, 2) 1/12 and Rodgers, 2009 includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Minimal Generating Pairs that are not (2, 3, r ) Small Strong Symmetric Genus Quotient (2, 3, r ) for r not relatively prime to 6 (2, 3, r ) for r Relatively Prime to 6 (2, 5, r ), (2, 6, 6), and (3, q, 4) And other minimal generating pairs that are not of the form (2, 3, r ). Min. Gen. Pair example group σQ (G) Reference Bowser, Partridge (2, 5, 5) AD(5, m) 1/20 and Rodgers, 2009 (2, 5, 6) Coxeter D6 2/30 J, 2007 (2, 5, 7) M22 11/140 Woldar, 1990 (2, 6, 6) Coxeter F4 1/12 J, 2007 (3, 3, 4) M24 1/24 Conder, 1991 (3, 4, 4) Coxeter D6 1/12 J, 2007 Notice that the triples (3, 3, 5) and (3, 3, 6) produce the same strong symmetric genus quotient as the triples (2, 5, 6) and (2, 6, 6), respectively. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Minimal Generating Pairs that are not (2, 3, r ) Small Strong Symmetric Genus Quotient (2, 3, r ) for r not relatively prime to 6 (2, 3, r ) for r Relatively Prime to 6 Using the groups D(3, m) and AD(4, m) We are left to show that for each r ≥ 7, there is a group where the strong symmetric genus is given by a (2, 3, r ) minimal generating pair. Recall the following results: When 3 m, D(3, m) has a minimal (2, 3, 2m) generating pair. When m is odd, AD(4, m) has a minimal (2, 3, 3m) generating pair. We will generalize the results for D(3, m) and AD(4, m) to include when m either 0 or 1. D(3, 1) = Σ3 which has a minimal (2, 2, 3) generating pair. AD(4, 1) = A4 which has a (2, 3, 3) generating pair D(3, 0) and AD(4, 0) are the trivial group. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Minimal Generating Pairs that are not (2, 3, r ) Small Strong Symmetric Genus Quotient (2, 3, r ) for r not relatively prime to 6 (2, 3, r ) for r Relatively Prime to 6 (2, 3, r ) generating pairs Recall the following results: When 3 m, D(3, m) has a minimal (2, 3, 2m) generating pair. When m is odd, AD(4, m) has a minimal (2, 3, 3m) generating pair. If m and p are both at least 1, such that 3 m, p is odd, and 2m is relatively prime to 3p, then D(3, m) × AD(4, p) has a minimal (2, 3, 6m · p) generating pair. These results reduce the open cases to (2, 3, r ) generating pairs with r relatively prime to 6. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Square Free r Small Strong Symmetric Genus Quotient General r (2, 3, r ) for r Relatively Prime to 6 Prime r Recall from work of Glover and Sjerve we saw that if r ≥ 7 is prime there exists a group PSL2 (p) with a minimal (2, 3, r ) generating pair We ﬁnd a prime p such that: p ≥ 13, p ±1 mod 8, p ±1 mod 9, p ±1 mod q for each prime q with 5 ≤ q < r , and p ≡ ±1 mod r Such a prime is exists by Dirichlet’s theorem on primes in arithmetic progressions. This prime p gives a PSL2 (p) with a minimal (2, 3, r ) generating pair. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Square Free r Small Strong Symmetric Genus Quotient General r (2, 3, r ) for r Relatively Prime to 6 Square Free r If r is not divisible by 2 or 3 and r is square free, then we can write r as a product of distinct primes qi for i = 1, 2, . . . n each at least 5. for each qi ≥ 7 we ﬁnd a prime pi such that pi ≥ 13, pi ±1 mod 8, pi ±1 mod 9, and pi ±1 mod s for each prime s qi such that 5 ≤ s < r . if qi = 5, then let pi = 5 n Then i=1 PSL2 (pi ) has a minimal (2, 3, r ) generating pair. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Square Free r Small Strong Symmetric Genus Quotient General r (2, 3, r ) for r Relatively Prime to 6 r a prime power Recall again from work of Glover and Sjerve, that if q ≥ 5 is a prime and n ≥ 2 with r = q n , there exists a prime p > 2 such that n−1 the group PSL2 (pq (q−1)/2 ) has a minimal (2, 3, r ) generating pair. p is chosen to be congruent modulo q n to a primitive root modulo qn. n−1 (q−1) Since p is congruent modulo q n to a primitive root, pq ≡1 mod q n . n−1 (q−1)/2 This means that pq ≡ −1 mod q n . n−1 (q−1)/2 So q n |pq + 1. And qn cannot divide pk + 1 or pk − 1 for any k < q n−1 (q − 1)/2. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Square Free r Small Strong Symmetric Genus Quotient General r (2, 3, r ) for r Relatively Prime to 6 General r If r is not divisible by 2 or 3, then we can write r as a product s which is square free and t which is the product of primes each to a power at least 2. As before we write s as a product of distinct primes qi for i = 1, 2, . . . n each at least 5. For each i there is a prime pi such that n PSL2 (pi ) has a i=1 minimal (2, 3, s) generating pair. k We write t as a product of mj j where the mj are distinct primes for j = 1, 2, . . . h. kj −1 mj (mj −1)/2 For each j there is a prime bj such that PSL2 bj has a k minimal (2, 3, mj j ) generating pair. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Square Free r Small Strong Symmetric Genus Quotient General r (2, 3, r ) for r Relatively Prime to 6 General r Each prime on the previous slide can be choosen so that n h kj −1 mj (mj −1)/2 PSL2 (pi ) × PSL2 j b i=1 j=1 has a minimal (2, 3, r ) generating pair. This shows that a for a general r a group can be constructed with a (2, 3, r ) minimal generating pair. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient Introduction Known Results Square Free r Small Strong Symmetric Genus Quotient General r (2, 3, r ) for r Relatively Prime to 6 Summary Theorem Given an integer r ≥ 7, there is a ﬁnite group G which has a minimal (2, 3, r ) generating pair. Notice that the group G can be found as the product of one or more of the groups D(3, m), AD(4, n), PSL2 (p), and/or PSL2 (q k ) for some integers m, n, and k as well as primes p and q. Corollary 1 Any strong symmetric genus quotient less than 12 , that is allowed by Singerman’s lemma, is the strong symmetric genus quotient of some ﬁnite group G. includegraphics[height=0.1 cm]GCClogo4.jpg M. Jackson - Grove City College Every possible small strong symmetric genus quotient

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