Burnside's basis theorem

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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


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


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

  • 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)