Burnside basis theorem

Theorem 1

If $G$ is a finite $p$-group, then $\mathrm{Frat}\mathit{}G\mathrm{=}{G}^{\mathrm{\prime }}\mathit{}{G}^p$, where $\mathrm{Frat}\mathit{}G$ is the Frattini subgroup, $G^{\mathrm{\prime }}$ the commutator subgroup, and $G^p$ is the subgroup generated by $p$-th powers.

The theorem implies that $G/\mathrm{Frat}G$ is elementary abelian, and thus has a minimal generating set of exactly $n$ elements, where $|G:\mathrm{Frat}G|=p^n$. Since any lift of such a generating set also generates $G$ (by the non-generating property of the Frattini subgroup), the smallest generating set of $G$ also has $n$ elements.

The theorem also holds for profinite $p$-groups (inverse limit of finite $p$-groups).

Title	Burnside basis theorem
Canonical name	BurnsideBasisTheorem
Date of creation	2013-03-22 13:16:08
Last modified on	2013-03-22 13:16:08
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	9
Author	alozano (2414)
Entry type	Theorem
Classification	msc 20D15