# proof of the Burnside basis theorem

Let $P$ be a $p$-group and $\mathrm{\Phi}(P)$ its Frattini subgroup^{}.

Every maximal subgroup $Q$ of $P$ is of index $p$ in $P$ and is therefore
normal in $P$. Thus $P/Q\cong {\mathbb{Z}}_{p}$. So given
$g\in P$, ${g}^{p}\in Q$
which proves ${P}^{p}\le Q$. Likewise, ${\mathbb{Z}}_{p}$ is abelian^{} so
$[P,P]\le Q$. As $Q$ is any maximal subgroup, it follows $[P,P]$ and
${P}^{p}$ lie in $\mathrm{\Phi}(P)$.

Now both $[P,P]$ and ${P}^{p}$ are characteristic subgroups of $P$ so in particular
$F=[P,P]{P}^{p}$ is normal in $P$. If we pass to $V=P/F$ we find that $V$ is abelian and every element has order $p$ – that is, $V$ is a vector space^{} over ${\mathbb{Z}}_{p}$. So the maximal subgroups of $P$ are in a 1-1 correspondence with the hyperplanes^{} of $V$. As the intersection^{} of all hyperplanes of a vector space is the origin, it follows the intersection of all maximal subgroups of $P$ is $F$. That is, $[P,P]{P}^{p}=\mathrm{\Phi}(P)$.

Title | proof of the Burnside basis theorem |
---|---|

Canonical name | ProofOfTheBurnsideBasisTheorem |

Date of creation | 2013-03-22 15:46:25 |

Last modified on | 2013-03-22 15:46:25 |

Owner | Algeboy (12884) |

Last modified by | Algeboy (12884) |

Numerical id | 12 |

Author | Algeboy (12884) |

Entry type | Proof |

Classification | msc 20D15 |