Burnside basis theorem
Theorem 1
If is a finite -group, then , where is the
Frattini subgroup, the commutator subgroup
, and is the subgroup
generated by -th powers.
The theorem implies that is elementary
abelian, and thus has a minimal generating set of
exactly elements, where . Since any lift of such a
generating set also generates (by the non-generating property of the
Frattini subgroup), the smallest generating set of also
has elements.
The theorem also holds for profinite -groups (inverse limit of finite -groups).
Title | Burnside basis theorem |
---|---|
Canonical name | BurnsideBasisTheorem |
Date of creation | 2013-03-22 13:16:08 |
Last modified on | 2013-03-22 13:16:08 |
Owner | alozano (2414) |
Last modified by | alozano (2414) |
Numerical id | 9 |
Author | alozano (2414) |
Entry type | Theorem |
Classification | msc 20D15 |