# Burnside's basis theorem

This article describes a fact or result that is not basic but it still well-established and standard. The fact may involve terms that are themselves non-basic
View other semi-basic facts in group theory
VIEW FACTS USING THIS:

## Statement

### Symbolic statement

Let  be a -group for some prime , and let  denote the Frattini subgroup of . Then,  is the largest elementary Abelian quotient of , and hence is a vector space over the prime field .

Burnside's basis theorem states that:

• A subset  of  is a generating set for  iff the image of  in  generates  as a -vector space.
• A subset  of  is a minimal generating set for  iff the image of  in  is a vector space basis for .

## 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 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.

## 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.