#### Vol. 13, No. 1, 2013

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A rank 3 geometry for the O'Nan group connected with the Livingstone graph

### Thomas Connor

Vol. 13 (2013), No. 1, 83–95
##### Abstract

We construct a rank $3$ geometry $\Gamma \left({O}^{\prime }N\right)$ over the diagram whose automorphism group is the O’Nan sporadic simple group. The maximal parabolic subgroups are the Janko group ${J}_{1}$, $2×{S}_{5}$ and the Mathieu group ${M}_{11}$. Our construction is based on a convenient amalgam of known geometries of rank $2$ for ${J}_{1}$ and ${M}_{11}$ extracted from the subgroup lattice of ${O}^{\prime }N$.

##### Keywords
Diagram geometry, coset geometry, sporadic simple groups, O'Nan group, Livingstone graph
##### Mathematical Subject Classification 2010
Primary: 20D08, 51E24