Burnside basis theorem

Theorem 1

If G is a finite p-group, then FratG=GGp, where FratG is the Frattini subgroupMathworldPlanetmath, G the commutator subgroupMathworldPlanetmath, and Gp is the subgroupMathworldPlanetmathPlanetmath generated by p-th powers.

The theorem implies that G/FratG is elementary abelian, and thus has a minimal generating setPlanetmathPlanetmath of exactly n elements, where |G:FratG|=pn. 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 limitMathworldPlanetmathPlanetmath 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