# Burnside basis theorem

###### Theorem 1

If $G$ is a finite $p$-group, then $\mathrm{Frat}\,G=G^{\prime}G^{p}$, where $\mathrm{Frat}\,G$ is the Frattini subgroup, $G^{\prime}$ 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 BurnsideBasisTheorem 2013-03-22 13:16:08 2013-03-22 13:16:08 alozano (2414) alozano (2414) 9 alozano (2414) Theorem msc 20D15