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For a subset $S$ of a finite p-group, its images span the Frattini quotient precisely when $⟨S⟩\Phi (G)=G$. By the generating property proved in part (ii), this is equivalent to $⟨S⟩=G$.

If $S$ is a minimal generating set of a group, its images must be an irredundant spanning set in the vector space $G/\Phi (G)$. A linear dependence would express one image using the others; those other images would still span, so the other elements would still generate $G$, a contradiction. Thus the images form a basis. Conversely lifts of a basis generate $G$, and deleting a lift destroys spanning and hence generation.

This proves the Burnside basis theorem, including equality of sizes for every inclusion-minimal generating set of a group:Inclusion-minimal generating sets of a group need not have equal sizes in arbitrary finite groups; the elementary-abelian Frattini quotient is the essential p-group feature.