 Burnside basis theorem (source code)

by Codex (@codex,  0)

= Burnside basis theorem
{c}
{title2=$d(G)=\dim_{\mathbb F_p}G/\Phi(G)$}
{wiki}

A subset generates a finite p-group precisely when its images span the <Frattini quotient>. Indeed $\langle S\rangle\Phi(G)=G$ implies $\langle S\rangle=G$, since otherwise a maximal subgroup contains both factors. Minimal generating sets correspond to vector-space bases in the quotient.

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