A New Construction for the First Janko Group

Horine, Thomas L. (2022) A New Construction for the First Janko Group. Doctoral thesis, UIN SAIZU Purwokerto.

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Abstract

In the mid 1960’s, Zvonomir Janko discovered a finite simple group, J1, of order 175560. [Ja65],[Ja66] His original formulation of the group was as the unique simple group with abelian 2-Sylow subgroups and an involution centralizer isomorphic to 2×A5. This first Janko group is exceptional, not only as one of the 26 sporadic finite simple groups, but also as one of only 6 pariahs (a sporadic finite simple group that does not appear as a subgroup or subquotient of the Monster). [Wi86] Other than the Matthieu groups, J1 is the smallest sporadic group. There exists a 266-point graph, called the Livingstone graph, whose automorphism group is J1. We study this graph in great depth, paying particular attention to how various maximal subgroups of J1 act on it. In each case, an attempt is made to find a simple structure within the graph that “explains” the subgroup. Upon completing this task, we use the discussion of the subgroup 2×A5 to build an orbifold comprised of 1463 dodecahedra whose symmetry group is exactly J1 (and a manifold whose symmetry group is 2 × J1). In addition, there is a final chapter discussing the various cyclogons of the Liv�ingstone graph. (A cyclogon of a graph is a sequence of vertices permuted by some graph automorphism.) Basic defintions are made, propositions proved, and a no�tion of equivalence is given, along with an exhaustive enumeration of the equivalence classes of cyclogons of various types.

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