Reading the figure
An arrow carries a marked source extension, including lifts and squares, into a scalar pushout. The square identity is preserved. An independently prescribed native marking needs its own comparison.
Mathematical labels
\(F_0(a)^2=F_0(a^2)\)
Selectable labels from the original PDF
Native PDF text retains the figure’s extraction order. Equations and data tables above provide the reading order.
Marked source extension finite subgroup + lifts + squares Marked image in scalar pushout F0(a) 2 = F0(a 2 ) The whole finite subgroup travels through the map. A separately fixed native marking needs its own comparison.
Original figure caption
Squares are preserved by the actual map of marked extensions.
For a full character \(\theta_i\), its scalar pushout is
A center-fixing comparison retains every supplied character value. In the dyadic case retain the image of the entire marked finite subgroup, its central embedding, lifts, squares, refinements, and phases. An actual extension map \(F_0\) obeys \(F_0(a)^2=F_0(a^2)\).
Scope
The unmarked complex pushouts of \(D_8\) and \(Q_8\) coincide, while their marked finite \(\mu_2\) extensions differ. Retaining only the central \(\{\pm1\}\) loses that distinction. Invisible kernel elements already removed by the central character cannot be recovered by pushout.