The book is a research monograph on the finite simple group known as the fourth Janko group. The existence of the group was discovered by Zvonimir Janko in 1976 and constructed by a group of authors (D. J. Benson, J. H. Conway, S. P. Norton, R. A. Parker and J. G. Thackakray) in Cambridge in 1980. The aim of the book is to provide a geometric characterization of the fourth Janko group. The group is characterized as the only group X acting transitively on incident vertex-edge pairs of a connected regular graph of valency 31 satisfying further local properties. The properties are encoded in the structure of the stabilizers in X of a vertex of and of an edge containing this vertex and in the way these two subgroups intersect. The union of the two stabilizers generates a subgroup Y of X. If the graph is a tree, then the subgroup is the free amalgamated product of the two stabilizers amalgamated over their intersection. It appears that there are only two further isomorphism types of the subgroup Y. These two types correspond in a highly nontrivial way to a complete bipartite graph or to the Petersen graph. The main theorem of the book states that the fourth Janko group corresponds to the Petersen graph. The book can be recommended to anyone interested in the geometries behind sporadic simple groups.

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