Burnside's basis theorem

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Statement

Symbolic statement

Let $P$ be a $p$-group for some prime $p$, and let $\mathrm{\Phi }(P)$ denote the Frattini subgroup of $P$. Then, $P/\mathrm{\Phi }(P)$ 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 $P/\mathrm{\Phi }(P)$ generates $P/\mathrm{\Phi }(P)$ 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 $P/\mathrm{\Phi }(P)$ is a vector space basis for $P/\mathrm{\Phi }(P)$.

Related facts

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.

A more general fact in group theory is:

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.

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.