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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

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              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



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                    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




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                     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


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                  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.



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                   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.
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                    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.
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                    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 ).

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                       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.



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                    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]
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                 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]


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                  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.




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                 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.




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                 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
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                      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.
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                    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.

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                    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.

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                    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.
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                     M. Jackson - Grove City College        Every possible small strong symmetric genus quotient

				
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