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 t2, u2, v2, (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, tutvtu. 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 Z2 × Z2.

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 Z2, D6, S4, Z2× S4 or Z2× A5, respectively.

If G or H is perfect and 1 d 4, then L = G × H.

G / G' H / H' Z2

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' Z2

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' Z2 and H / H' Z4

[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' Z4

[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 J1 and J2 such that

[L : J1] > 6, [L : J2] > 6

and J1 J1 = 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, P48 of order 48.

P48 u, v | u3 = v2 = (uv)3(u-1v)3 = 1 P48 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 P48