Babai-Beals filtration
From Groupprops
Definition
Let be a finite group. The Babai-Beals filtration of is the following ascending chain of subgroups of :
where:
- denotes the trivial subgroup of .
- denotes the solvable radical of .
- denotes the socle over solvable radical of , i.e., .
- denotes the permutation kernel of .
Here is what we can say about the successive quotients:
- is a solvable group.
- is a direct product of simple non-abelian groups (with some of them possibly isomorphic).
- is isomorphic to a subgroup of the external direct product of the outer automorphism groups of some simple non-abelian groups (the same ones that appear in the quotient , with the same repetitions). Hence, by Schreier's conjecture (which has been proved conditional to the classification of finite simple groups) it is a solvable group.
- has order at most , hence is relatively small compared to the size of .
References
- A polyomial-time theory of black-box groups I by Laszlo Babai and Robert Beals, : ^{Ungated copy (PDF)}^{More info}