Nilpotent residual

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

Definition

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

1. It is the intersection of all nilpotent-quotient normal subgroups of the group (i.e., normal subgroups such that the quotient is a nilpotent group).
2. It is the intersection of all members of the (finite) 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.