# Lower Fitting series

## Definition

Let $G$ be a finite group. The lower Fitting series of $G$ is a descending subgroup series defined as follows:

• The zeroth member is $G$.
• The $i^{th}$ member is the nilpotent residual, or equivalently, the hypocenter, of the $(i-1)^{th}$ member -- it is the intersection of all nilpotent-quotient normal subgroups of its predecessor.

The series reaches the trivial subgroup if and only if $G$ is solvable, i.e., is a finite solvable group. The lower Fitting series of a finite solvable group is the fastest descending Fitting series for the group. In particular, the length of this series is the Fitting length of the group.

In general, for a finite possibly non-solvable group, the lower Fitting series stabilizes at the solvable residual of the group, which coincides with the perfect core (because we are dealing with a finite group).

## Related notions

• Derived series: This is obtained by iterating the derived subgroup operator. Derived series is to abelian groups what lower Fitting series is to nilpotent groups.
• Lower central series: In fact, each member of the lower Fitting series is the hypocenter of its predecessor, which is defined as the endpoint of the lower central series of its predecessor.
• Upper Fitting series: This is the fastest ascending Fitting series for a finite group and is obtained by quotient-iteration on the Fitting subgroup.