Subdirect Products of M* Groups Coy L. May and Jay Zimmerman We are interested in groups acting as motions of compact surfaces, with and without boundary.

Restrictions on the Order A compact surface with genus g  2 has at most 84(g – 1) automorphisms by Hurwitz Theorem. If only automorphisms which preserve the orientation of the surface are considered, then the bound becomes 42(g – 1).

Bordered Klein Surfaces A compact bordered Klein surface of genus g  2 has at most 12(g – 1) automorphisms. A bordered surface for which the bound is attained is said to have maximal symmetry and its group is called an M* group.

M* group properties Let Γ be the group generated by t, u and v, with relators t 2, u 2, v 2, (tu) 2, (tv) 3. A finite group G is an M* group if and only if G is the image of Γ. If G is an M* group, the order of the element uv is called an action index of G and is denoted q = o(uv).

Fundamental result G is the automorphism group of a bordered Klein surface X with maximal symmetry and k boundary components, where |G| = 2qk iff G is an M* group. Each component of the boundary  X is fixed by a dihedral subgroup of G of order 2q.

Canonical Subgroups of G G + =  tu, uv  and G' =  tv, tu  tv  tu . G' ≤ G + ≤ G, where each subgroup has index 1 or 2 in the larger group. X is orientable iff [G : G + ] = 2. G/G' is the image of Z 2 × Z 2.

Subdirect Product Let G and H be M* groups. So  : Γ  G and  : Γ  H. Define  : Γ  G × H by  (x) = (  (x),  (x)). L = Im(  ) is a subdirect product and an M* group.

Normal Subgroup of G Define  G (  ) =  (ker(  )) and  H (  ) =  (ker(  )).  G (  ) is a normal subgroup of G.  H (  ) is a normal subgroup of H.  G (  ) × {1} = Im(  )  (G × {1})

Index of the subdirect product |G /  G (  )| = [G × H : L] = |H /  H (  )|. G /  G (  )  Γ/(ker(  )  ker(  ))  H /  H (  ).

Obvious Consequences Suppose that H is a simple group. Then  H (  ) is either {1} or H. If  H (  ) = 1, then G /  G (  )  H. If  H (  ) = H, then L = G × H.

Action Indices Let G and H be M* groups with action indices q and r and let d = gcd(q, r). For 1  d  5, then G /  G (  ) is the image of Z 2, D 6, S 4, Z 2 × S 4 or Z 2 × A 5, respectively. If G or H is perfect and 1  d  4, then L = G × H.

G / G'  H / H'  Z 2 Let G and H be M* groups If ker(  )  ker(  )  Γ', then [G × H : L]  2. If ker(  )  ker(  )  Γ', then G /  G (  ) is perfect.

G / G'  H / H'  Z 2 Suppose that the only quotients of G and H that are isomorphic are abelian. If ker(  )  ker(  )  Γ', then [G × H : L] = 2. If ker(  )  ker(  )  Γ', then L = G × H.

G / G'  Z 2 and H / H'  Z 4 [G × H : L]  2. Suppose that the only quotients of G and H that are isomorphic are abelian. [G × H : L] = 2.

G / G'  H / H'  Z 4 [G × H : L]  4. Suppose that the only quotients of G and H that are isomorphic are abelian. [G × H : L] = 4.

Necessary Conditions The M* group L is a subdirect product of two smaller M* groups iff L has normal subgroups J 1 and J 2 such that [L : J 1 ] > 6, [L : J 2 ] > 6 and J 1  J 1 = 1.

Corollary Let L be an M* group with |L| > 12 and its Fitting subgroup F(L) divisible by two prime numbers. Then L is a subdirect product of two smaller M* groups.

Conclusion These techniques can be used with many different maximal actions, such as Hurwitz groups, odd order groups acting maximally on Riemann surfaces, p-groups acting similarly. Finally, I would like to draw some group actions on Riemann surfaces.

Burnside Burnside 1911 talked about actions on compact surfaces. He even gave a picture of the action of the Quaternion Group on a surface of Genus 2.

Quaternion Group Properties The surface has genus 2 and 16 region. Each vertex has degree 8, corresponding to a rotation of order 4. Image of Triangle Group, T(4,4,4). Highly symmetric.

Dicyclic Group of Order 12

Quasiabelian Group of Order 16

Orientation Reversing Actions Suppose that G acts on a surface with orientation reversing elements and G + is the image of a triangle group. Therefore, G is the image of either a Full Triangle group or of a Hybrid Triangle group.

The group, P 48 of order 48. P 48   u, v | u 3 = v 2 = (uv) 3 (u -1 v) 3 = 1  P 48 has symmetric genus 2. It is the image of HT(3,4) which is a subgroup of FT(3,8,2). The hyperbolic space region is distorted into a polygonal region.

Polygonal Representation of P 48