# 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 |