Frattini Geometry of Non-Generating Complexes: Symmetry, Homology and Betti Numbers

For a finite group \(G\), let \(N(G)\) be the simplicial complex of subsets that do not generate \(G\). We separate a general Frattini-topological reduction from the structure special to finite \(p\)-groups. For every finite noncyclic group, the non-cone Frattini core is homotopy equivalent to the o