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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.
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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.
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.
