Nilpotent residual

Definition
The nilpotent residual of a group can be defined in the following equivalent ways:


 * 1) It is the intersection of all defining ingredient::nilpotent-quotient normal subgroups of the group (i.e., defining ingredient::normal subgroups such that the quotient is a defining ingredient::nilpotent group).
 * 2) It is the intersection of all members of the (finite) defining ingredient::lower central series of the group. In particular, it is the $$\omega^{th}$$ member of the transfinite lower central series.

A group is residually nilpotent if and only if its nilpotent residual is trivial.

When the group is a finite group or a virtually nilpotent group, and in some other nice situations, the nilpotent residual of a group itself is a nilpotent-quotient normal subgroup. If this is the case, it can also be described as the unique smallest nilpotent-quotient normal subgroup.

The nilpotent residual of a group $$G$$ is denoted $$\gamma_\omega(G)$$ or $$\gamma_\infty(G)$$.

Related notions

 * Hypocenter is the ultimate stable member of the transfinite lower central series. For a finite group or a virtually nilpotent group, the hypocenter coincides with the nilpotent residual.
 * Lower Fitting series for a finite group is a Fitting series where each successive member is the nilpotent residual of its predecessor.