Burnside's basis theorem: Difference between revisions
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===Symbolic statement=== | ===Symbolic statement=== | ||
Let <math>P</math> be a <math>p</math>-group for some prime <math>p</math>, and let <math>\Phi(P)</math> denote the [[Frattini subgroup]] of <math>P</math>. Then, <math>P/\Phi(P)</math> is the largest elementary | Let <math>P</math> be a <math>p</math>-group for some prime <math>p</math>, and let <math>\Phi(P)</math> denote the [[Frattini subgroup]] of <math>P</math>. Then, <math>P/\Phi(P)</math> is the largest elementary abelian quotient of <math>P</math>, and hence is a vector space over the prime field <math>\mathbb{F}_p</math>. | ||
Burnside's basis theorem states that: | Burnside's basis theorem states that: | ||
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* A subset <math>S</math> of <math>P</math> is a [[generating set]] for <math>P</math> iff the image of <math>S</math> in <math>P/\Phi(P)</math> generates <math>P/\Phi(P)</math> as a <math>\mathbb{F}_p</math>-vector space. | * A subset <math>S</math> of <math>P</math> is a [[generating set]] for <math>P</math> iff the image of <math>S</math> in <math>P/\Phi(P)</math> generates <math>P/\Phi(P)</math> as a <math>\mathbb{F}_p</math>-vector space. | ||
* A subset <math>S</math> of <math>P</math> is a [[minimal generating set]] for <math>P</math> iff the image of <math>S</math> in <math>P/\Phi(P)</math> is a vector space basis for <math>P/\Phi(P)</math>. | * A subset <math>S</math> of <math>P</math> is a [[minimal generating set]] for <math>P</math> iff the image of <math>S</math> in <math>P/\Phi(P)</math> is a vector space basis for <math>P/\Phi(P)</math>. | ||
==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 [[cal:Nakayama's lemma|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== | ==Proof== | ||
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* 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 | * 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. | * Any element in the Frattini subgroup can be dropped from any generating set. | ||
==References== | |||
===Textbook references=== | |||
* {{booklink-stated|DummitFoote}}, Exercise 26(a), Page 199 (Section 6.2) | |||
==See also== | |||
* For other theorems called "Burnside's theorem", see [[Burnside's theorem]] for disambiguation. | |||
Latest revision as of 10:58, 23 June 2024
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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 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)
See also
- For other theorems called "Burnside's theorem", see Burnside's theorem for disambiguation.