Burnside's basis theorem
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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 .
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
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.
- 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)