Burnside's basis theorem

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Statement

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 $P/\Phi (P)$ .

Related facts

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 generated, 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.

References

Textbook references

Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Exercise 26(a), Page 199 (Section 6.2)