proof of the Burnside basis theorem

Let P be a p-group and Φ(P) its Frattini subgroupMathworldPlanetmath.

Every maximal subgroup Q of P is of index p in P and is therefore normal in P. Thus P/Qp. So given gP, gpQ which proves PpQ. Likewise, p is abelianMathworldPlanetmath so [P,P]Q. As Q is any maximal subgroup, it follows [P,P] and Pp lie in Φ(P).

Now both [P,P] and Pp are characteristic subgroups of P so in particular F=[P,P]Pp 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 spaceMathworldPlanetmath over p. So the maximal subgroups of P are in a 1-1 correspondence with the hyperplanesMathworldPlanetmathPlanetmath of V. As the intersectionMathworldPlanetmath 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]Pp=Φ(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