Fitting’s theorem
Fitting’s Theorem states that if is a group
and and are normal nilpotent subgroups (http://planetmath.org/Subgroup) of ,
then is also a normal nilpotent subgroup
(of nilpotency class less than or equal to
the sum of the nilpotency classes of and ).
Thus, any finite group has a unique largest normal nilpotent subgroup, called its Fitting subgroup
.
More generally, the Fitting subgroup of a group is defined to be the subgroup of generated by the normal nilpotent subgroups of ;
Fitting’s Theorem shows that the Fitting subgroup is always locally nilpotent.
A group that is equal to its own Fitting subgroup is sometimes called a Fitting group.
Title | Fitting’s theorem |
---|---|
Canonical name | FittingsTheorem |
Date of creation | 2013-03-22 13:51:39 |
Last modified on | 2013-03-22 13:51:39 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 12 |
Author | yark (2760) |
Entry type | Theorem |
Classification | msc 20D25 |
Defines | Fitting subgroup |
Defines | Fitting group |