Burnside's basis theorem

Symbolic statement
Let $$P$$ be a $$p$$-group for some prime $$p$$, and let $$\Phi(P)$$ denote the Frattini subgroup of $$P$$. Then, $$P/\Phi(P)$$ is the largest elementary Abelian quotient of $$P$$, and hence is a vector space over the prime field $$\mathbb{F}_p$$.

Burnside's basis theorem states that:


 * A subset $$S$$ of $$P$$ is a generating set for $$P$$ iff the image of $$S$$ in $$P/\Phi(P)$$ generates $$P/\Phi(P)$$ as a $$\mathbb{F}_p$$-vector space.
 * A subset $$S$$ of $$P$$ is a minimal generating set for $$P$$ iff the image of $$S$$ in $$P/\Phi(P)$$ is a vector space basis for $$P/\Phi(P)$$.

Generalizations
Frattini subgroup is finitely generated implies subset is generating set iff image in Frattini quotient is: If the Frattini subgroup of any group is finitely gnerated, then a subset of the whole group is a generating set iff its image mod the Frattini subgroup is a generating set for the Frattini quotient.

Burnside's basis theorem closely parallels certain formulations, and corollaries, of Nakayama's lemma, which states that generating sets for a module are in correspondence with generating sets for its top, which is its quotient by its Jacobson radical. Here, the Jacobson radical of a module plays the role of the Frattini subgroup, as the set of nongenerators.

Related fact on p-groups
Burnside's theorem on coprime automorphisms and Frattini subgroup

Proof
The proof follows directly from the following two facts:


 * Combining a generating set for a normal subgroup, and a set of inverse images (via the quotient map) of the quotient group gives us a generating set for the quotient group
 * Any element in the Frattini subgroup can be dropped from any generating set.

Textbook references

 * , Exercise 26(a), Page 199 (Section 6.2)