Graphical Representation of Groups of Genus Two and Their Hyperelliptic Curves (original) (raw)

Abstract

DESCRIPTION In this paper, we investigate distinct types of groups of genus two in complete detail. We prove that, there are exactly four types of groups of genus two and we give their presentations as finitely presented, transitive permutation representations of certain degrees. We give their character tables, matrix representations, and their primary invariants and their Cayley color graphs and/or their Shreier’s coset graphs. We also compute the equations of the hyperelliptic curves covered by these four types of groups, which happen to be the groups obtained from the maximal automorphism bound for the soluble (|G|=48(g-1)), supersoluble |G|=24, nilpotent (16(g-1)) and the quaternion (4, 4, 4)-group of order 8 for g = 2.

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