= Generator number and first cohomology of a finitely generated pro-p group
{title2=$d(G)=\dim_{\mathbb F_p}H^1(G,\mathbb F_p)$}
For a finitely generated <pro-p group>, its <profinite Frattini subgroup> has finite <elementary abelian> quotient. Continuous <group homomorphisms> to $\mathbb F_p$ are precisely the linear functionals on this quotient. The topological generation criterion identifies a minimal generating set with a basis, giving the displayed equality. Finite generation is essential when generator number means the least cardinality of an unrestricted dense generating family.
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