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Dowling geometry
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<p>In <a href="/facts/Combinatorics/k14xUoKT">combinatorial</a> <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a Dowling geometry, named after Thomas A. Dowling, is a <a href="/facts/Matroid/5bLcGlRF">matroid</a> associated with a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a>. There is a Dowling geometry of each rank for each group. If the rank is at least 3, the Dowling geometry uniquely determines the group. Dowling geometries have a role in matroid theory as <a href="/facts/Universal_object/oeckvXRj">universal objects</a> (Kahn and Kung, 1982); in that respect they are analogous to <a href="/facts/Projective_geometry/rHKSaEbx">projective geometries</a>, but based on groups instead of <a href="/facts/Field_(mathematics)/xAjAS4ko">fields</a>. </p><p>A Dowling lattice is the <a href="/facts/Geometric_lattice/DkCTIQL1">geometric lattice</a> of <a href="/facts/Matroid/5bLcGlRF">flats</a> associated with a Dowling geometry. The lattice and the geometry are mathematically equivalent: knowing either one determines the other. Dowling lattices, and by implication Dowling geometries, were introduced by Dowling (1973a,b). </p><p>A Dowling lattice or geometry of rank <i>n</i> of a group <i>G</i> is often denoted by <i>Qn</i>(<i>G</i>). </p>